title

Maths for Programmers Tutorial - Full Course on Sets and Logic

description

Learn the maths and logic concepts that are important for programmers to understand.
Shawn Grooms explains the following concepts:
⌨️ (00:00) Tips For Learning
⌨️ (01:32) What Is Discrete Mathematics?
⌨️ (03:45) Sets - What Is A Set?
⌨️ (06:22) Sets - Interval Notation & Common Sets
⌨️ (08:25) Sets - What Is A Rational Number?
⌨️ (10:18) Sets - Here Is A Non-Rational Number
⌨️ (12:17) Sets - Set Operators
⌨️ (13:45) Sets - Set Operators (Examples)
⌨️ (15:49) Sets - Subsets & Supersets
⌨️ (17:30) Sets - The Universe & Complements
⌨️ (20:02) Sets - Subsets & Supersets (Examples)
⌨️ (21:56) Sets - The Universe & Complements (Examples)
⌨️ (24:16) Sets - Idempotent & Identity Laws
⌨️ (25:14) Sets - Complement & Involution Laws
⌨️ (27:08) Sets - Associative & Commutative Laws
⌨️ (28:42) Sets - Distributive Law (Diagrams)
⌨️ (30:22) Sets - Distributive Law Proof (Case 1)
⌨️ (32:07) Sets - Distributive Law Proof (Case 2)
⌨️ (33:48) Sets - Distributive Law (Examples)
⌨️ (35:25) Sets - DeMorgan’s Law
⌨️ (37:32) Sets - DeMorgan’s Law (Examples)
⌨️ (39:38) Logic - What Is Logic?
⌨️ (41:26) Logic - Propositions
⌨️ (43:06) Logic - Composite Propositions
⌨️ (44:41) Logic - Truth Tables
⌨️ (46:30) Logic - Idempotent & Identity Laws
⌨️ (48:13) Logic - Complement & Involution Laws
⌨️ (49:58) Logic - Commutative Laws
⌨️ (51:35) Logic - Associative & Distributive Laws
⌨️ (53:09) Logic - DeMorgan’s Laws
⌨️ (54:23) Logic - Conditional Statements
⌨️ (55:45) Logic - Logical Quantifiers
⌨️ (57:59) Logic - What Are Tautologies?
--
Learn to code for free and get a developer job: https://www.freecodecamp.org
Read hundreds of articles on programming: https://medium.freecodecamp.org

detail

{'title': 'Maths for Programmers Tutorial - Full Course on Sets and Logic', 'heatmap': [{'end': 697.987, 'start': 508.939, 'weight': 1}, {'end': 836.547, 'start': 715.196, 'weight': 0.784}], 'summary': 'This full course on sets and logic covers discrete mathematics tips for programmers, set notation, rational and irrational numbers, set operators, algebra laws, and the importance of logic in mathematics with detailed explanations and examples.', 'chapters': [{'end': 215.069, 'segs': [{'end': 122.888, 'src': 'embed', 'start': 4.48, 'weight': 0, 'content': [{'end': 10.561, 'text': 'Hello world, Sean Grooms here with Free Code Camp and in this video I will be giving you three tips on how to learn discrete mathematics.', 'start': 4.48, 'duration': 6.081}, {'end': 13.162, 'text': 'The first tip is to stay calm.', 'start': 11.201, 'duration': 1.961}, {'end': 21.943, 'text': "People often hear the word mathematics and panic and when you panic you stop listening and if you stop listening you can't learn the material.", 'start': 14.022, 'duration': 7.921}, {'end': 27.024, 'text': 'So if you simply stay calm then you will be able to learn the material better.', 'start': 22.464, 'duration': 4.56}, {'end': 30.025, 'text': "You aren't being graded on the subject so there's no need to panic.", 'start': 27.544, 'duration': 2.481}, {'end': 32.924, 'text': 'The second is to rewind.', 'start': 31.263, 'duration': 1.661}, {'end': 35.846, 'text': 'You can rewind the videos at any time.', 'start': 34.065, 'duration': 1.781}, {'end': 39.748, 'text': "And when I learned discrete mathematics, I didn't have this opportunity.", 'start': 36.706, 'duration': 3.042}, {'end': 45.452, 'text': 'So in theory, you should be able to learn the material much faster than I was ever able to.', 'start': 40.229, 'duration': 5.223}, {'end': 48.794, 'text': 'So I strongly encourage rewinding videos.', 'start': 45.592, 'duration': 3.202}, {'end': 51.035, 'text': 'Finally, explain.', 'start': 49.815, 'duration': 1.22}, {'end': 59.761, 'text': 'Once you think you understand the material of the videos, you can try to explain the material out loud either to yourself or to a friend.', 'start': 51.876, 'duration': 7.885}, {'end': 66.123, 'text': "If no one's around or you feel weird talking to yourself, then you can try to explain it to rubber ducky.", 'start': 60.256, 'duration': 5.867}, {'end': 75.135, 'text': "There's a concept in programming called rubber duck debugging, which is where programmers go through their code line by line,", 'start': 66.744, 'duration': 8.391}, {'end': 79.4, 'text': 'trying to identify bugs in their code by explaining it to rubber duck.', 'start': 75.135, 'duration': 4.265}, {'end': 85.614, 'text': 'So. similarly, if you go through the material of the videos, subject by subject,', 'start': 80.071, 'duration': 5.543}, {'end': 91.037, 'text': 'you will soon be able to identify any gaps in your understanding of the material.', 'start': 85.614, 'duration': 5.423}, {'end': 101.903, 'text': "In this video, I'll be explaining what discrete mathematics is and why it's important for the field of computer science and programming.", 'start': 96, 'duration': 5.903}, {'end': 108.005, 'text': 'Discrete mathematics is a branch of mathematics that deals with discrete or finite sets of elements,', 'start': 102.944, 'duration': 5.061}, {'end': 110.246, 'text': 'rather than continuous or infinite sets of elements.', 'start': 108.005, 'duration': 2.241}, {'end': 115.847, 'text': 'Imagine trying to run a program that requires an infinite number of executions to complete a task.', 'start': 111.246, 'duration': 4.601}, {'end': 122.888, 'text': "It's obvious to say that the program would run forever and the task would never be completed because there are an infinite number of executions.", 'start': 116.647, 'duration': 6.241}], 'summary': 'Tips for learning discrete mathematics: stay calm, rewind videos, explain material out loud.', 'duration': 118.408, 'max_score': 4.48, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI4480.jpg'}], 'start': 4.48, 'title': 'Learning discrete mathematics', 'summary': 'Provides tips for learning discrete mathematics, emphasizing staying calm, utilizing video rewind, and explaining material to enhance comprehension. it also explains the importance of discrete mathematics in computer science, focusing on finite sets approximating infinite sets and its impact on programming.', 'chapters': [{'end': 51.035, 'start': 4.48, 'title': 'Tips for learning discrete mathematics', 'summary': 'Provides three tips for learning discrete mathematics, emphasizing the importance of staying calm, utilizing the ability to rewind videos, and explaining the material, which can enhance learning and comprehension.', 'duration': 46.555, 'highlights': ['Staying calm is emphasized as it enables better learning, and it is highlighted that individuals are not being graded on the subject, reducing the need to panic.', 'The ability to rewind videos is encouraged as it allows for faster learning, with the speaker mentioning that this opportunity was unavailable during their own learning experience.', 'The importance of explaining the material for enhanced comprehension is mentioned as a final tip for effective learning.']}, {'end': 215.069, 'start': 51.876, 'title': 'Importance of discrete mathematics', 'summary': 'Explains the concept of discrete mathematics and its relevance to computer science, emphasizing the use of finite sets to approximate infinite sets and its impact on programming.', 'duration': 163.193, 'highlights': ['The concept of discrete mathematics is explained, emphasizing its importance for computer science and programming.', 'Discrete mathematics deals with discrete or finite sets of elements, rather than continuous or infinite sets, impacting the completion of tasks in programming.', 'The use of finite sets to approximate infinite sets is illustrated with the example of a circle and the physical impossibility of completing the task with an infinite number of points.', 'The explanation of how regular polygons can be used as approximations for circles by increasing the number of vertices is provided, highlighting the practical application of discrete mathematics in computer graphics.']}], 'duration': 210.589, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI4480.jpg', 'highlights': ['The importance of discrete mathematics in computer science and programming', 'Staying calm is emphasized for better learning, not being graded on the subject', 'Utilizing video rewind for faster learning and improved comprehension', 'Explaining material to enhance comprehension and effective learning', 'Discrete mathematics deals with finite sets approximating infinite sets', 'Practical application of discrete mathematics in computer graphics']}, {'end': 492.254, 'segs': [{'end': 291.53, 'src': 'embed', 'start': 242.76, 'weight': 3, 'content': [{'end': 247.883, 'text': 'And for example, elements could be numbers, letters, variables, more sets, or nothing at all.', 'start': 242.76, 'duration': 5.123}, {'end': 256.538, 'text': 'Sets are usually denoted by capital letters such as capital A, capital B, C, E, or F.', 'start': 249.152, 'duration': 7.386}, {'end': 259.641, 'text': 'And we say that the set contains elements.', 'start': 256.538, 'duration': 3.103}, {'end': 266.488, 'text': 'For example, we could say that the set A equals the set containing the element 5.', 'start': 259.801, 'duration': 6.687}, {'end': 269.01, 'text': 'or we can say that five is an element of A.', 'start': 266.488, 'duration': 2.522}, {'end': 271.933, 'text': 'Now, I will be introducing a lot of symbols like this.', 'start': 269.01, 'duration': 2.923}, {'end': 277.618, 'text': "This just means is an element of, and it'll save us a lot of time in future videos.", 'start': 272.153, 'duration': 5.465}, {'end': 282.101, 'text': "I don't have to write out is an element of each time, so that'll come in handy.", 'start': 278.238, 'duration': 3.863}, {'end': 285.044, 'text': 'So what does a set actually look like?', 'start': 283.403, 'duration': 1.641}, {'end': 291.53, 'text': "Well, there are two common types of notation for sets, and the one I'm introducing in this video is roster notation,", 'start': 285.084, 'duration': 6.446}], 'summary': 'Introducing sets, elements, and roster notation in mathematics.', 'duration': 48.77, 'max_score': 242.76, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI242760.jpg'}, {'end': 372.371, 'src': 'embed', 'start': 342.612, 'weight': 0, 'content': [{'end': 347.816, 'text': 'And finally, the set F equals the set containing elements A, B, and C.', 'start': 342.612, 'duration': 5.204}, {'end': 357.463, 'text': 'In this case our elements are sets, so we would read this as F equals the set containing the element 5, the set containing elements 2, 3, and 4,', 'start': 347.816, 'duration': 9.647}, {'end': 359.484, 'text': 'and the set containing D, F and G.', 'start': 357.463, 'duration': 2.021}, {'end': 372.371, 'text': "So in the next video, I'll be going over a different form of set notation in addition to introducing you to some common sets.", 'start': 363.985, 'duration': 8.386}], 'summary': 'Introducing set f with elements a, b, c, 5, 2, 3, 4, d, f, g.', 'duration': 29.759, 'max_score': 342.612, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI342612.jpg'}, {'end': 426.376, 'src': 'embed', 'start': 395.297, 'weight': 1, 'content': [{'end': 401.42, 'text': 'The beauty of interval notation is that it allows us to efficiently describe all numbers between two values.', 'start': 395.297, 'duration': 6.123}, {'end': 407.342, 'text': 'Suppose we wanted to say that the variable x is between 0 and 1.', 'start': 402.1, 'duration': 5.242}, {'end': 408.983, 'text': 'Well, we could do just that.', 'start': 407.342, 'duration': 1.641}, {'end': 418.45, 'text': 'We could say that x is an element of the interval 0 to 1, which is really saying x is greater than 0 and x is less than 1.', 'start': 409.163, 'duration': 9.287}, {'end': 426.376, 'text': 'Likewise, if we wanted to say that x is greater than or equal to 0 and x is less than 1, then we have to switch this to a square bracket.', 'start': 418.45, 'duration': 7.926}], 'summary': 'Interval notation efficiently describes numbers between two values using square brackets and parentheses.', 'duration': 31.079, 'max_score': 395.297, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI395297.jpg'}, {'end': 467.879, 'src': 'embed', 'start': 442.287, 'weight': 2, 'content': [{'end': 448.051, 'text': "And now I'll talk about some common sets, the first one being the null set, which is the same thing as the empty set.", 'start': 442.287, 'duration': 5.764}, {'end': 455.496, 'text': 'which is just there are no elements contained within the set, and this will come in handy in the future.', 'start': 448.675, 'duration': 6.821}, {'end': 466.019, 'text': 'The funny N here is the natural numbers, which is the set containing elements 1,, 2, 3, and then these three dots here are called an ellipsis,', 'start': 456.597, 'duration': 9.422}, {'end': 467.879, 'text': 'which tells us that the pattern continues.', 'start': 466.019, 'duration': 1.86}], 'summary': 'Discussed null set and natural numbers in sets.', 'duration': 25.592, 'max_score': 442.287, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI442287.jpg'}], 'start': 215.069, 'title': 'Sets and notation', 'summary': 'Introduces sets and roster notation, explaining the representation of sets using symbols, the concept of elements, and the utility of roster notation. it also covers interval notation for efficiently describing numbers and setting the foundation for understanding algorithms.', 'chapters': [{'end': 291.53, 'start': 215.069, 'title': 'Sets and roster notation', 'summary': 'Discusses the concept of sets, including their representation using roster notation and the use of symbols to denote elements. it explains that sets are collections of distinct objects and introduces the concept of elements and their representation using capital letters and symbols. it also emphasizes the utility of roster notation for representing sets.', 'duration': 76.461, 'highlights': ['The chapter discusses the concept of sets, including their representation using roster notation and the use of symbols to denote elements.', 'It explains that sets are collections of distinct objects and introduces the concept of elements and their representation using capital letters and symbols.', 'It also emphasizes the utility of roster notation for representing sets.']}, {'end': 492.254, 'start': 291.53, 'title': 'Introduction to sets and interval notation', 'summary': 'Introduces the concept of sets, including set notation and common sets, and also explains interval notation for efficiently describing numbers between two values, while setting the foundation for understanding algorithms.', 'duration': 200.724, 'highlights': ['The chapter introduces the concept of sets and set notation, explaining the notation for different sets A, B, C, E, and F, with examples of elements and their representations.', 'The chapter discusses interval notation and its efficiency in describing numbers between two values, with examples of using interval notation to define ranges for a variable x.', 'The chapter explains common sets such as the null set, natural numbers, and whole numbers, providing examples and descriptions for each set.']}], 'duration': 277.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI215069.jpg', 'highlights': ['The chapter introduces the concept of sets and set notation, explaining the notation for different sets A, B, C, E, and F, with examples of elements and their representations.', 'The chapter discusses interval notation and its efficiency in describing numbers between two values, with examples of using interval notation to define ranges for a variable x.', 'It explains common sets such as the null set, natural numbers, and whole numbers, providing examples and descriptions for each set.', 'The chapter discusses the concept of sets, including their representation using roster notation and the use of symbols to denote elements.', 'It also emphasizes the utility of roster notation for representing sets.', 'It explains that sets are collections of distinct objects and introduces the concept of elements and their representation using capital letters and symbols.']}, {'end': 726.018, 'segs': [{'end': 697.987, 'src': 'heatmap', 'start': 492.257, 'weight': 1, 'content': [{'end': 499.299, 'text': 'And finally, we have all of the integers positive and negative and non-negative, which is 0, negative, 1, negative, 2,', 'start': 492.257, 'duration': 7.042}, {'end': 504.3, 'text': 'all the way down to negative infinity and 0,, 1,, 2,, 3, 4, all the way up to infinity.', 'start': 499.299, 'duration': 5.001}, {'end': 512.822, 'text': "In this video, I'll be introducing the rational numbers and set-builder notation.", 'start': 508.939, 'duration': 3.883}, {'end': 517.547, 'text': 'The rational numbers are defined as a ratio of two integers.', 'start': 513.703, 'duration': 3.844}, {'end': 524.092, 'text': 'And examples include the integers themselves, terminating decimals, and repeating decimals.', 'start': 518.587, 'duration': 5.505}, {'end': 526.414, 'text': "So let's start with repeating decimals.", 'start': 524.493, 'duration': 1.921}, {'end': 534.281, 'text': 'If we let 10x equal 9.999 repeating, then if we divide both sides by 10, then x equals 0.999 repeating.', 'start': 526.735, 'duration': 7.546}, {'end': 540.063, 'text': 'And if we subtract x from 10x, we have 9x equals 9.', 'start': 535.762, 'duration': 4.301}, {'end': 543.444, 'text': 'And then divide both sides by 9, we have x equals 1.', 'start': 540.063, 'duration': 3.381}, {'end': 548.225, 'text': 'And then if we multiply both sides by 10, we have 10x equals 10.', 'start': 543.444, 'duration': 4.781}, {'end': 553.146, 'text': 'So now we have 10x equals 10, but 10x also equals 9.99 and repeating.', 'start': 548.225, 'duration': 4.921}, {'end': 558.243, 'text': 'So we have now, and these are in fact true, 9.99 repeating equals 10.', 'start': 553.626, 'duration': 4.617}, {'end': 565.386, 'text': 'So we can now express 10x as an integer, which is a rational number.', 'start': 558.243, 'duration': 7.143}, {'end': 569.207, 'text': 'We can put 10 divided by 1, which is a ratio.', 'start': 565.466, 'duration': 3.741}, {'end': 574.209, 'text': 'So another problem we have is terminating decimals.', 'start': 570.308, 'duration': 3.901}, {'end': 581.632, 'text': 'If we have 2.78 equal to Y, then we can just easily say 278 divided by 100 equals Y as well.', 'start': 574.727, 'duration': 6.905}, {'end': 586.855, 'text': "These are two integers, and it's another rational number.", 'start': 582.392, 'duration': 4.463}, {'end': 595.561, 'text': 'So a convenient way to define these is denoted with a Q, and this is called set builder notation.', 'start': 587.355, 'duration': 8.206}, {'end': 604.247, 'text': 'We have the Q equals the set containing elements A divided by B such that A and B are elements of the integers.', 'start': 595.601, 'duration': 8.646}, {'end': 607.828, 'text': 'with the only condition that b cannot equal 0.', 'start': 604.927, 'duration': 2.901}, {'end': 611.849, 'text': "If you divide by 0, it's undefined.", 'start': 607.828, 'duration': 4.021}, {'end': 612.87, 'text': "It doesn't mean anything.", 'start': 611.869, 'duration': 1.001}, {'end': 616.211, 'text': "So if you're interested in that, I recommend googling it.", 'start': 613.01, 'duration': 3.201}, {'end': 617.131, 'text': "I think it's interesting.", 'start': 616.231, 'duration': 0.9}, {'end': 624.954, 'text': "In this video, I'll be giving an example of a non-rational number.", 'start': 621.673, 'duration': 3.281}, {'end': 631.936, 'text': 'If the square root of 2 were a rational number, then by definition, it must be expressed as the ratio of two integers.', 'start': 625.814, 'duration': 6.122}, {'end': 637.173, 'text': 'If we then square both sides, we have 2 equals a squared divided by b squared.', 'start': 632.99, 'duration': 4.183}, {'end': 642.517, 'text': 'If we multiply both sides by b squared, we have 2b squared equals a squared.', 'start': 639.034, 'duration': 3.483}, {'end': 644.879, 'text': 'And from this, we know that a squared is even.', 'start': 643.057, 'duration': 1.822}, {'end': 650.062, 'text': 'a squared is even because we have 2 times some integer, which is b squared.', 'start': 645.939, 'duration': 4.123}, {'end': 656.867, 'text': 'Since a squared is even, we know that a is even because a squared equals 2 times some integer.', 'start': 651.183, 'duration': 5.684}, {'end': 658.969, 'text': 'The integer I chose was 2k squared.', 'start': 656.967, 'duration': 2.002}, {'end': 665.331, 'text': 'that means that a times a can be expressed as 2k times 2k which is to say that a equals 2k.', 'start': 659.847, 'duration': 5.484}, {'end': 676.36, 'text': 'I can now substitute 2 times 2k squared for a squared and that leaves us with b squared equals 2k squared once these twos cancel out.', 'start': 666.132, 'duration': 10.228}, {'end': 681.984, 'text': 'Now, this also means that b squared is even because b squared equals 2 times some integer,', 'start': 677, 'duration': 4.984}, {'end': 685.507, 'text': 'which is to say that b is also even following this same logic.', 'start': 681.984, 'duration': 3.523}, {'end': 697.987, 'text': 'Now if we go back to our original premise The square root of 2 equals a divided by b, where a divided by b is an irreducible fraction.', 'start': 686.487, 'duration': 11.5}], 'summary': 'Introduction to rational numbers and set-builder notation, including examples and conditions, with a brief mention of non-rational numbers.', 'duration': 25.29, 'max_score': 492.257, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI492257.jpg'}, {'end': 604.247, 'src': 'embed', 'start': 543.444, 'weight': 2, 'content': [{'end': 548.225, 'text': 'And then if we multiply both sides by 10, we have 10x equals 10.', 'start': 543.444, 'duration': 4.781}, {'end': 553.146, 'text': 'So now we have 10x equals 10, but 10x also equals 9.99 and repeating.', 'start': 548.225, 'duration': 4.921}, {'end': 558.243, 'text': 'So we have now, and these are in fact true, 9.99 repeating equals 10.', 'start': 553.626, 'duration': 4.617}, {'end': 565.386, 'text': 'So we can now express 10x as an integer, which is a rational number.', 'start': 558.243, 'duration': 7.143}, {'end': 569.207, 'text': 'We can put 10 divided by 1, which is a ratio.', 'start': 565.466, 'duration': 3.741}, {'end': 574.209, 'text': 'So another problem we have is terminating decimals.', 'start': 570.308, 'duration': 3.901}, {'end': 581.632, 'text': 'If we have 2.78 equal to Y, then we can just easily say 278 divided by 100 equals Y as well.', 'start': 574.727, 'duration': 6.905}, {'end': 586.855, 'text': "These are two integers, and it's another rational number.", 'start': 582.392, 'duration': 4.463}, {'end': 595.561, 'text': 'So a convenient way to define these is denoted with a Q, and this is called set builder notation.', 'start': 587.355, 'duration': 8.206}, {'end': 604.247, 'text': 'We have the Q equals the set containing elements A divided by B such that A and B are elements of the integers.', 'start': 595.601, 'duration': 8.646}], 'summary': 'Explaining rational numbers: 9.99 repeating equals 10, 2.78 equals 278 divided by 100.', 'duration': 60.803, 'max_score': 543.444, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI543444.jpg'}, {'end': 715.196, 'src': 'embed', 'start': 681.984, 'weight': 0, 'content': [{'end': 685.507, 'text': 'which is to say that b is also even following this same logic.', 'start': 681.984, 'duration': 3.523}, {'end': 697.987, 'text': 'Now if we go back to our original premise The square root of 2 equals a divided by b, where a divided by b is an irreducible fraction.', 'start': 686.487, 'duration': 11.5}, {'end': 707.894, 'text': 'But since a is an even number and b is also an even number, we can express these numbers as 2k and 2l.', 'start': 699.352, 'duration': 8.542}, {'end': 715.196, 'text': "But these twos here will reduce, meaning that we'll have the square root of 2 equals k divided by l.", 'start': 708.394, 'duration': 6.802}], 'summary': 'Using the logic, the square root of 2 equals k divided by l.', 'duration': 33.212, 'max_score': 681.984, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI681984.jpg'}], 'start': 492.257, 'title': 'Rational numbers and irrationality', 'summary': 'Introduces rational numbers as a ratio of two integers, including repeating and terminating decimals, and the set-builder notation. it also demonstrates the proof that the square root of 2 is irrational by showing that both a and b are even numbers, resulting in the reduced form of a/b as k/l, leading to the conclusion that the square root of 2 is not a rational number.', 'chapters': [{'end': 631.936, 'start': 492.257, 'title': 'Introduction to rational numbers', 'summary': 'Introduces rational numbers as a ratio of two integers, including repeating and terminating decimals, and the set-builder notation for defining rational numbers.', 'duration': 139.679, 'highlights': ['The rational numbers are defined as a ratio of two integers, including integers, terminating decimals, and repeating decimals.', 'Repeating decimals like 9.999 repeating can be expressed as a ratio, such as 10x equals 9.999 repeating, leading to x equals 1, making it a rational number.', 'Terminating decimals like 2.78 can also be expressed as a ratio, such as 278 divided by 100, making it a rational number.', 'The set-builder notation Q equals the set containing elements A divided by B such that A and B are elements of the integers, with the only condition that B cannot equal 0.', 'The chapter gives an example of a non-rational number by considering the square root of 2, which cannot be expressed as a ratio of two integers.']}, {'end': 726.018, 'start': 632.99, 'title': 'Square root of 2 proof', 'summary': 'Demonstrates a proof that the square root of 2 is irrational by showing that both a and b are even numbers, resulting in the reduced form of a/b as k/l, leading to the conclusion that the square root of 2 is not a rational number.', 'duration': 93.028, 'highlights': ['The chapter demonstrates a proof that the square root of 2 is irrational by showing that both a and b are even numbers, resulting in the reduced form of a/b as k/l, leading to the conclusion that the square root of 2 is not a rational number.', 'Both a and b are even numbers, as a squared and b squared are even, implying that a and b themselves are also even.', 'The reduced form of a/b as k/l leads to the conclusion that the square root of 2 is not a rational number.']}], 'duration': 233.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI492257.jpg', 'highlights': ['The chapter demonstrates a proof that the square root of 2 is irrational by showing that both a and b are even numbers, resulting in the reduced form of a/b as k/l, leading to the conclusion that the square root of 2 is not a rational number.', 'The rational numbers are defined as a ratio of two integers, including integers, terminating decimals, and repeating decimals.', 'The set-builder notation Q equals the set containing elements A divided by B such that A and B are elements of the integers, with the only condition that B cannot equal 0.', 'Repeating decimals like 9.999 repeating can be expressed as a ratio, such as 10x equals 9.999 repeating, leading to x equals 1, making it a rational number.', 'Terminating decimals like 2.78 can also be expressed as a ratio, such as 278 divided by 100, making it a rational number.']}, {'end': 1234.116, 'segs': [{'end': 761.056, 'src': 'embed', 'start': 726.977, 'weight': 0, 'content': [{'end': 735.739, 'text': "Now there's a more specific way to define these numbers or classify these numbers, but we need set operators in the next video.", 'start': 726.977, 'duration': 8.762}, {'end': 744.021, 'text': "In this video, I'll be defining four binary operators for sets.", 'start': 740.46, 'duration': 3.561}, {'end': 749.942, 'text': "I've defined two sets A and B to help give some examples.", 'start': 744.941, 'duration': 5.001}, {'end': 761.056, 'text': 'So the first is A union B, which equals the set containing elements P such that P is an element of A or P is an element of B.', 'start': 751.007, 'duration': 10.049}], 'summary': 'Defining four binary operators for sets, using sets a and b as examples.', 'duration': 34.079, 'max_score': 726.977, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI726977.jpg'}, {'end': 844.113, 'src': 'embed', 'start': 813.974, 'weight': 4, 'content': [{'end': 824.607, 'text': 'So the irrational numbers are simply the real numbers minus the rational numbers, and that equals the irrational numbers.', 'start': 813.974, 'duration': 10.633}, {'end': 836.547, 'text': 'In this video, I will be using Venn diagrams to give a graphical representation of the set operators we learned in the previous video.', 'start': 829.301, 'duration': 7.246}, {'end': 844.113, 'text': 'So, if you recall the definition of A union B, you know that A union B equals the set containing elements x,', 'start': 837.607, 'duration': 6.506}], 'summary': 'Irrational numbers are real numbers minus rational numbers.', 'duration': 30.139, 'max_score': 813.974, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI813974.jpg'}, {'end': 974.806, 'src': 'embed', 'start': 942.797, 'weight': 2, 'content': [{'end': 947.579, 'text': 'You should really focus on the definitions as it will help you out in the long term.', 'start': 942.797, 'duration': 4.782}, {'end': 955.692, 'text': "In this video, I'll be introducing the concept of subsets and supersets.", 'start': 952.25, 'duration': 3.442}, {'end': 960.776, 'text': 'The dots on the board here represent elements of a set.', 'start': 956.913, 'duration': 3.863}, {'end': 964.959, 'text': 'The lines represent the sets themselves.', 'start': 962.397, 'duration': 2.562}, {'end': 969.282, 'text': 'These lines have been labeled B, A, and U.', 'start': 965.72, 'duration': 3.562}, {'end': 974.806, 'text': 'We can say that B is a subset of A because all elements of B are elements of A.', 'start': 969.282, 'duration': 5.524}], 'summary': 'Introduction to subsets and supersets with clear examples and labels.', 'duration': 32.009, 'max_score': 942.797, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI942797.jpg'}, {'end': 1065.125, 'src': 'embed', 'start': 1037.916, 'weight': 5, 'content': [{'end': 1043.28, 'text': 'So U is a subset of A, and A is actually a subset of U.', 'start': 1037.916, 'duration': 5.364}, {'end': 1046.281, 'text': 'U is a superset of A, and A is also a superset of U.', 'start': 1043.28, 'duration': 3.001}, {'end': 1049.604, 'text': 'So the sets A and U in this case are actually equal.', 'start': 1046.281, 'duration': 3.323}, {'end': 1058.85, 'text': "In this video, I'll be introducing the concept of the universal set and complements.", 'start': 1054.587, 'duration': 4.263}, {'end': 1065.125, 'text': 'So I want to begin by talking about the universal set, which is commonly referred to as the universe.', 'start': 1059.864, 'duration': 5.261}], 'summary': 'Introduction to the concept of the universal set and complements.', 'duration': 27.209, 'max_score': 1037.916, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1037915.jpg'}, {'end': 1218.032, 'src': 'embed', 'start': 1182.035, 'weight': 3, 'content': [{'end': 1189.623, 'text': 'So if this is one and two, or better yet, how about this? This is the rational numbers and these are the integers.', 'start': 1182.035, 'duration': 7.588}, {'end': 1200.866, 'text': 'So everything outside of this would be the irrational numbers and the non-integer values.', 'start': 1190.443, 'duration': 10.423}, {'end': 1209.17, 'text': "In this video, I'll be giving examples for subsets and supersets.", 'start': 1205.629, 'duration': 3.541}, {'end': 1213.911, 'text': 'So if you look at this blue line here, this represents our universal set.', 'start': 1209.77, 'duration': 4.141}, {'end': 1218.032, 'text': "We've defined it to be the integers from 1 to 11.", 'start': 1215.231, 'duration': 2.801}], 'summary': 'The video discusses rational and irrational numbers using integers from 1 to 11 as the universal set.', 'duration': 35.997, 'max_score': 1182.035, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1182035.jpg'}], 'start': 726.977, 'title': 'Set operators and set relationships', 'summary': 'Introduces four binary operators for sets: union, intersection, set difference, and symmetric difference, along with their graphical representations using venn diagrams. it also explains subsets, supersets, and universal sets, and the relationships between sets, using integers from 1 to 11 as the universal set.', 'chapters': [{'end': 942.357, 'start': 726.977, 'title': 'Set operators and venn diagrams', 'summary': 'Introduces four binary operators for sets: union, intersection, set difference, and symmetric difference, and illustrates their graphical representations using venn diagrams, providing a clear understanding of the concept of set operators.', 'duration': 215.38, 'highlights': ['The chapter introduces four binary operators for sets: union, intersection, set difference, and symmetric difference, providing a clear understanding of the concept of set operators.', 'The video uses Venn diagrams to give a graphical representation of the set operators, aiding in visualizing the concept of set operations.', 'The concept of irrational numbers is defined as the real numbers minus the rational numbers.']}, {'end': 1234.116, 'start': 942.797, 'title': 'Subsets, supersets, and universal set', 'summary': 'Introduces the concepts of subsets, supersets, and universal set, explaining the relationships between sets, the concept of complements, and providing examples, using the integers from 1 to 11 as the universal set.', 'duration': 291.319, 'highlights': ['The chapter explains the relationship between subsets and supersets, emphasizing the concept of proper subsets and proper supersets.', 'It discusses the concept of the universal set and the importance of defining it, highlighting that it represents the maximum boundaries of a set and clarifying the concept of complements.', 'The chapter provides examples of subsets and supersets, using the integers from 1 to 11 as the universal set, and clarifies that elements outside the universe are not within the set in question.']}], 'duration': 507.139, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI726977.jpg', 'highlights': ['The chapter introduces four binary operators for sets: union, intersection, set difference, and symmetric difference, providing a clear understanding of the concept of set operators.', 'The video uses Venn diagrams to give a graphical representation of the set operators, aiding in visualizing the concept of set operations.', 'The chapter explains the relationship between subsets and supersets, emphasizing the concept of proper subsets and proper supersets.', 'The chapter provides examples of subsets and supersets, using the integers from 1 to 11 as the universal set, and clarifies that elements outside the universe are not within the set in question.', 'The concept of irrational numbers is defined as the real numbers minus the rational numbers.', 'It discusses the concept of the universal set and the importance of defining it, highlighting that it represents the maximum boundaries of a set and clarifying the concept of complements.']}, {'end': 1603.061, 'segs': [{'end': 1356.54, 'src': 'embed', 'start': 1309.966, 'weight': 4, 'content': [{'end': 1315.631, 'text': 'So if we wanted to be more specific, we could actually say that C is a proper subset of the universe.', 'start': 1309.966, 'duration': 5.665}, {'end': 1322.558, 'text': "In this video, I'll be going over examples of complements.", 'start': 1320.236, 'duration': 2.322}, {'end': 1331.126, 'text': "So in this video, I've defined the universe to be the integers 1 through 5, and it's denoted graphically by this blue line.", 'start': 1323.118, 'duration': 8.008}, {'end': 1341.262, 'text': 'And if we recall the definition of the complements with regards to A, we have A complement equals the set containing elements X,', 'start': 1332.119, 'duration': 9.143}, {'end': 1346.844, 'text': 'such that X is an element of the universe and X is not an element of A.', 'start': 1341.262, 'duration': 5.582}, {'end': 1356.54, 'text': 'So in that case, we can look at all the integers, 1 through 5, and if it contains any element from A, we throw it out.', 'start': 1346.844, 'duration': 9.696}], 'summary': 'Defining complements and subsets within the universe of integers 1 through 5.', 'duration': 46.574, 'max_score': 1309.966, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1309966.jpg'}, {'end': 1409.829, 'src': 'embed', 'start': 1378.148, 'weight': 6, 'content': [{'end': 1387.011, 'text': "Well, we know that 1, 2, 4, and 5 are within the universe and they're not within B.", 'start': 1378.148, 'duration': 8.863}, {'end': 1389.811, 'text': 'And lastly, we have C complement.', 'start': 1387.011, 'duration': 2.8}, {'end': 1400.554, 'text': "2, 4, and 5 are within the universe and they're not within C because we had to exclude 1 and 3.", 'start': 1389.831, 'duration': 10.723}, {'end': 1409.829, 'text': "I want to begin this video by reviewing our definitions of the union and intersection After that, I'll be introducing two algebraic laws for sets.", 'start': 1400.554, 'duration': 9.275}], 'summary': 'Reviewing definitions of union and intersection, introducing two algebraic laws for sets.', 'duration': 31.681, 'max_score': 1378.148, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1378148.jpg'}, {'end': 1578.617, 'src': 'embed', 'start': 1490.097, 'weight': 0, 'content': [{'end': 1495.079, 'text': 'such that x is an element of A or x is an element of the null set.', 'start': 1490.097, 'duration': 4.982}, {'end': 1499.801, 'text': "But since we know that the null set is empty, we can't possibly have any elements in there.", 'start': 1495.379, 'duration': 4.422}, {'end': 1509.885, 'text': "So it's really saying that A union the null set is equal to the set containing elements x such that x is an element of A.", 'start': 1500.561, 'duration': 9.324}, {'end': 1513.007, 'text': 'Or more simply, it equals A.', 'start': 1509.885, 'duration': 3.122}, {'end': 1522.514, 'text': 'I want to begin this video by reviewing the definition of complements.', 'start': 1518.191, 'duration': 4.323}, {'end': 1531.641, 'text': 'So the definition of a complement of a set, for instance the complement of A, equals the set containing elements x,', 'start': 1523.575, 'duration': 8.066}, {'end': 1535.604, 'text': 'such that x is not an element of A.', 'start': 1531.641, 'duration': 3.963}, {'end': 1539.607, 'text': 'I will now be going over two new algebraic laws for sets.', 'start': 1535.604, 'duration': 4.003}, {'end': 1542.49, 'text': 'The first are the law of complements.', 'start': 1540.328, 'duration': 2.162}, {'end': 1549.317, 'text': 'The law of complements states that if we take the union of a set and its complement, it equals the universe.', 'start': 1543.291, 'duration': 6.026}, {'end': 1553.702, 'text': 'Or, if you take the complement of the null set, it equals the universe.', 'start': 1550.138, 'duration': 3.564}, {'end': 1559.942, 'text': 'Furthermore, the intersection of a set and its complement is the null set.', 'start': 1554.738, 'duration': 5.204}, {'end': 1569.99, 'text': 'And that should be clear, because if you have the set A and you are intersecting it with everything that is not inside A,', 'start': 1560.342, 'duration': 9.648}, {'end': 1572.272, 'text': "there's clearly not going to be any overlap.", 'start': 1569.99, 'duration': 2.282}, {'end': 1574.434, 'text': "So you're going to have an empty set.", 'start': 1572.632, 'duration': 1.802}, {'end': 1578.617, 'text': 'The next one is the law of involution.', 'start': 1576.615, 'duration': 2.002}], 'summary': 'Review of set theory, complements, and algebraic laws for sets.', 'duration': 88.52, 'max_score': 1490.097, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1490097.jpg'}], 'start': 1234.516, 'title': 'Sets and set algebra', 'summary': 'Discusses sets, subsets, complements, and set algebra laws, covering examples of subsets and complements and defining union, intersection, impotence law, identity laws, and new algebraic laws for sets.', 'chapters': [{'end': 1409.829, 'start': 1234.516, 'title': 'Sets: subsets, supersets, complements', 'summary': 'Discusses sets, subsets, and complements, with examples of sets being subsets and proper subsets of the universe, and the definition and examples of complements for sets a, b, and c in the universe of integers 1 through 5.', 'duration': 175.313, 'highlights': ['The chapter discusses examples of sets being subsets and proper subsets of the universe.', 'The definition and examples of complements for sets A, B, and C in the universe of integers 1 through 5 are provided, with specific elements included and excluded.', 'Reviewing the definitions of the union and intersection of sets is mentioned, followed by the introduction of two algebraic laws for sets.']}, {'end': 1603.061, 'start': 1409.849, 'title': 'Set algebra: laws and definitions', 'summary': 'Covers the definition of union and intersection of sets, impotence law, identity laws, complements, and new algebraic laws for sets including the law of complements and the law of involution.', 'duration': 193.212, 'highlights': ['The law of complements states that if we take the union of a set and its complement, it equals the universe.', 'The intersection of a set and its complement is the null set.', 'A union null set equals A.', 'The complement of the null set equals the universe.', 'The definition of a complement of a set equals the set containing elements x, such that x is not an element of A.']}], 'duration': 368.545, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1234516.jpg', 'highlights': ['The law of complements states that if we take the union of a set and its complement, it equals the universe.', 'The intersection of a set and its complement is the null set.', 'A union null set equals A.', 'The complement of the null set equals the universe.', 'The chapter discusses examples of sets being subsets and proper subsets of the universe.', 'The definition and examples of complements for sets A, B, and C in the universe of integers 1 through 5 are provided, with specific elements included and excluded.', 'Reviewing the definitions of the union and intersection of sets is mentioned, followed by the introduction of two algebraic laws for sets.', 'The definition of a complement of a set equals the set containing elements x, such that x is not an element of A.']}, {'end': 2367.659, 'segs': [{'end': 1756.905, 'src': 'embed', 'start': 1731.846, 'weight': 0, 'content': [{'end': 1739.553, 'text': 'The distributive law is simply saying that the set and operation will be distributed over another set and operation.', 'start': 1731.846, 'duration': 7.707}, {'end': 1747.617, 'text': 'For instance, A intersection B union C equals A intersection B union A intersection C.', 'start': 1740.154, 'duration': 7.463}, {'end': 1756.905, 'text': "So I thought it might be helpful to use Venn diagrams for the conceptual idea of this and then I'll do a separate video on the actual definition.", 'start': 1747.617, 'duration': 9.288}], 'summary': 'Distributive law: a intersection b union c equals a intersection b union a intersection c.', 'duration': 25.059, 'max_score': 1731.846, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1731846.jpg'}, {'end': 2044.814, 'src': 'embed', 'start': 2016.781, 'weight': 3, 'content': [{'end': 2026.827, 'text': "and we've shown that both sides of the equation are subsets of each other, and therefore the distributive law is in fact true.", 'start': 2016.781, 'duration': 10.046}, {'end': 2027.328, 'text': "it's proven.", 'start': 2026.827, 'duration': 0.501}, {'end': 2035.312, 'text': "In this video, I'll be going over an example of the distributive law.", 'start': 2031.991, 'duration': 3.321}, {'end': 2044.814, 'text': 'So if you look at A intersection B union C, that is supposed to equal A intersection B union A intersection C.', 'start': 2036.412, 'duration': 8.402}], 'summary': 'The distributive law is proven true with examples.', 'duration': 28.033, 'max_score': 2016.781, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2016781.jpg'}, {'end': 2177.086, 'src': 'embed', 'start': 2149.321, 'weight': 2, 'content': [{'end': 2163.82, 'text': "So we're going to prove that the complement of the union of A and B is in fact equal to the complement of A intersected with the complement of B.", 'start': 2149.321, 'duration': 14.499}, {'end': 2170.743, 'text': 'So to do this, we have to show that the complement of A union B is a subset of A complement intersection B complement and vice versa.', 'start': 2163.82, 'duration': 6.923}, {'end': 2172.504, 'text': "So let's get started.", 'start': 2171.844, 'duration': 0.66}, {'end': 2177.086, 'text': 'Suppose that x is an element of A union B complement.', 'start': 2173.544, 'duration': 3.542}], 'summary': 'Proving complement of union of a and b equals complement of a intersection complement of b.', 'duration': 27.765, 'max_score': 2149.321, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2149321.jpg'}, {'end': 2283.484, 'src': 'embed', 'start': 2262.03, 'weight': 1, 'content': [{'end': 2271.757, 'text': "De Morgan's Law states that the complement of the intersection equals the union of the complements and that the complement of the union equals the intersection of the complements.", 'start': 2262.03, 'duration': 9.727}, {'end': 2278.461, 'text': 'So, if we check out this new universal set here, it is equal to the set containing integers,', 'start': 2272.277, 'duration': 6.184}, {'end': 2283.484, 'text': 'such that x is greater than or equal to 0 and x is less than or equal to 5..', 'start': 2278.461, 'duration': 5.023}], 'summary': "De morgan's law relates complements and set operations.", 'duration': 21.454, 'max_score': 2262.03, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2262030.jpg'}], 'start': 1603.601, 'title': 'Set algebra laws and proofs', 'summary': "Introduces algebraic laws for sets such as associativity, commutativity, distributive law, and de morgan's law with detailed proofs and examples. it also explains de morgan's law using an example and proves the complement of the union of a and b is equal to the complement of a intersected with the complement of b, with a detailed example and logical proof.", 'chapters': [{'end': 2148.98, 'start': 1603.601, 'title': 'Set algebra laws and proofs', 'summary': "Introduces algebraic laws for sets, including associativity, commutativity, distributive law, and de morgan's law, and provides proofs and examples for each law in detail.", 'duration': 545.379, 'highlights': ['The distributive law states A intersection B union C equals A intersection B union A intersection C.', "De Morgan's Law states the complement of a union is the intersection of the complements, or the complement of the intersection is the union of the complements.", 'The proof shows that A intersection B union C is a subset of A intersection B union A intersection C, and A intersection B union A intersection C is a subset of A intersection B union C.', 'Examples are provided to demonstrate the application of the distributive law, showing the equality of A intersection B union C and A intersection B union A intersection C.']}, {'end': 2367.659, 'start': 2149.321, 'title': "De morgan's law and set complements", 'summary': "Explains de morgan's law using an example and proves that the complement of the union of a and b is equal to the complement of a intersected with the complement of b, with a detailed example and logical proof.", 'duration': 218.338, 'highlights': ['The complement of A union B is a subset of A complement intersection B complement, and vice versa, proven through logical deduction.', "De Morgan's Law is explained, stating that the complement of the intersection equals the union of the complements and that the complement of the union equals the intersection of the complements.", "Detailed example of De Morgan's Law is provided, showcasing the complement of A union B and A intersection B, along with their respective complements, proving their equality."]}], 'duration': 764.058, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI1603601.jpg', 'highlights': ['The distributive law states A intersection B union C equals A intersection B union A intersection C.', "De Morgan's Law states the complement of a union is the intersection of the complements, or the complement of the intersection is the union of the complements.", 'The complement of A union B is a subset of A complement intersection B complement, and vice versa, proven through logical deduction.', 'Examples are provided to demonstrate the application of the distributive law, showing the equality of A intersection B union C and A intersection B union A intersection C.']}, {'end': 3607.261, 'segs': [{'end': 2446.548, 'src': 'embed', 'start': 2415.741, 'weight': 0, 'content': [{'end': 2429.247, 'text': "we wouldn't be able to deduce or move from point A to point B and or make claims from point A to point B in a hundred percent affirmative,", 'start': 2415.741, 'duration': 13.506}, {'end': 2432.342, 'text': 'no questions asked way.', 'start': 2429.247, 'duration': 3.095}, {'end': 2435.363, 'text': "So that's very important for mathematics.", 'start': 2432.882, 'duration': 2.481}, {'end': 2440.465, 'text': "If you couldn't do that in mathematics, then it would all fall apart.", 'start': 2435.403, 'duration': 5.062}, {'end': 2446.548, 'text': 'There would be no certainty, there would be no foundation, no argument for us to stand on,', 'start': 2440.725, 'duration': 5.823}], 'summary': 'Certainty and foundation in mathematics are crucial for making claims and deductions.', 'duration': 30.807, 'max_score': 2415.741, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2415741.jpg'}, {'end': 2504.739, 'src': 'embed', 'start': 2474.299, 'weight': 1, 'content': [{'end': 2476.06, 'text': 'Well, programmers write code, first of all.', 'start': 2474.299, 'duration': 1.761}, {'end': 2477.4, 'text': 'Code uses algorithms.', 'start': 2476.38, 'duration': 1.02}, {'end': 2480.62, 'text': 'Algorithms is essentially just math.', 'start': 2477.52, 'duration': 3.1}, {'end': 2482.781, 'text': 'And mathematics requires logic.', 'start': 2481.16, 'duration': 1.621}, {'end': 2483.921, 'text': "So it's that simple.", 'start': 2483.101, 'duration': 0.82}, {'end': 2491.523, 'text': "In this video, I'll be introducing the notion of a proposition.", 'start': 2488.882, 'duration': 2.641}, {'end': 2497.247, 'text': 'A proposition is simply a declarative statement with a verifiable truth value.', 'start': 2491.543, 'duration': 5.704}, {'end': 2500.516, 'text': "they're usually denoted by lowercase letters.", 'start': 2498.035, 'duration': 2.481}, {'end': 2504.739, 'text': 'so if we have this lowercase, p equals, rain falls from the sky.', 'start': 2500.516, 'duration': 4.223}], 'summary': 'Programmers write code, which uses algorithms and logic. introducing propositions in this video.', 'duration': 30.44, 'max_score': 2474.299, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2474299.jpg'}, {'end': 2616.215, 'src': 'embed', 'start': 2588.482, 'weight': 8, 'content': [{'end': 2592.005, 'text': "In this video, I'll be introducing the concept of composite propositions.", 'start': 2588.482, 'duration': 3.523}, {'end': 2599.512, 'text': 'Just like regular propositions, composite propositions are declarative statements with a verifiable truth value.', 'start': 2592.706, 'duration': 6.806}, {'end': 2609.781, 'text': "Composite propositions are made up of sub-propositions, and in this video we're going to talk about the conjunction and the disjunction.", 'start': 2600.793, 'duration': 8.988}, {'end': 2616.215, 'text': 'the first one I want to introduce is the conjunction that is P and Q.', 'start': 2611.232, 'duration': 4.983}], 'summary': 'Introducing composite propositions with conjunction and disjunction.', 'duration': 27.733, 'max_score': 2588.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2588482.jpg'}, {'end': 2810.471, 'src': 'embed', 'start': 2779.22, 'weight': 4, 'content': [{'end': 2782.343, 'text': 'And these are by the definitions of the conjunction P.', 'start': 2779.22, 'duration': 3.123}, {'end': 2784.201, 'text': 'and the disjunction.', 'start': 2782.739, 'duration': 1.462}, {'end': 2787.985, 'text': "That's why these values take on true or false.", 'start': 2784.541, 'duration': 3.444}, {'end': 2797.897, 'text': "In this video, I'll introduce our first two algebraic laws of logic, idempotence and identities.", 'start': 2792.991, 'duration': 4.906}, {'end': 2800.18, 'text': 'These are the same laws we saw in set theory.', 'start': 2798.337, 'duration': 1.843}, {'end': 2810.471, 'text': 'If you recall, idempotence is when you can take a proposition and apply a binary operator to that proposition over and over,', 'start': 2801.468, 'duration': 9.003}], 'summary': 'Introduction to idempotence and identities in logic.', 'duration': 31.251, 'max_score': 2779.22, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2779220.jpg'}, {'end': 3128.627, 'src': 'embed', 'start': 3097.711, 'weight': 5, 'content': [{'end': 3101.912, 'text': "In this video, I'll be introducing the associative law and the distributive law for logic.", 'start': 3097.711, 'duration': 4.201}, {'end': 3107.414, 'text': "Now previously, we've only dealt with two primitive propositions.", 'start': 3102.852, 'duration': 4.562}, {'end': 3113.876, 'text': "So that's why we've only ever had four possible outcomes in our true or false path.", 'start': 3107.994, 'duration': 5.882}, {'end': 3117.977, 'text': "Now we're going to take this a step further and we're going to have eight.", 'start': 3114.716, 'duration': 3.261}, {'end': 3128.627, 'text': "propositions, or eight possible outcomes for our true or false path, because we're dealing with three propositions, P, Q, and R.", 'start': 3119.161, 'duration': 9.466}], 'summary': 'Introducing associative and distributive laws for logic, expanding from 4 to 8 possible outcomes.', 'duration': 30.916, 'max_score': 3097.711, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI3097711.jpg'}, {'end': 3272.361, 'src': 'embed', 'start': 3236.678, 'weight': 3, 'content': [{'end': 3238.499, 'text': 'So these two columns are equal.', 'start': 3236.678, 'duration': 1.821}, {'end': 3242.982, 'text': 'So we can definitively say that these are logical equivalents.', 'start': 3238.599, 'duration': 4.383}, {'end': 3255.032, 'text': 'Also, not P or Q is in fact not P and not Q based on our truth table.', 'start': 3244.884, 'duration': 10.148}, {'end': 3256.533, 'text': "It's 100% certain.", 'start': 3255.112, 'duration': 1.421}, {'end': 3261.297, 'text': "This is logic, and there's no argument against it.", 'start': 3256.693, 'duration': 4.604}, {'end': 3268.535, 'text': "In this video, I'll be introducing the notion of conditional statements.", 'start': 3265.751, 'duration': 2.784}, {'end': 3272.361, 'text': 'A conditional statement contains a hypothesis and a conclusion.', 'start': 3269.176, 'duration': 3.185}], 'summary': 'Logical equivalents of not p or q and not p and not q are introduced with 100% certainty.', 'duration': 35.683, 'max_score': 3236.678, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI3236678.jpg'}, {'end': 3543.38, 'src': 'embed', 'start': 3492.55, 'weight': 6, 'content': [{'end': 3500.672, 'text': 'So for here, we have p or not p, so true or false, true or false, false or true, or false or true, which yields nothing but a true column.', 'start': 3492.55, 'duration': 8.122}, {'end': 3504.893, 'text': 'Next we have the law of contradiction.', 'start': 3501.61, 'duration': 3.283}, {'end': 3510.878, 'text': 'So if you ignore this negation here in this column of trues here, we have P and not P.', 'start': 3505.273, 'duration': 5.605}, {'end': 3518.129, 'text': 'So true and false yields false, true and false yields false, false and true, false and true, all yield false.', 'start': 3512.165, 'duration': 5.964}, {'end': 3524.654, 'text': 'Now when we actually apply this negation to the false column here, that yields a column of truths.', 'start': 3518.63, 'duration': 6.024}, {'end': 3526.856, 'text': 'So the law of contradiction is a tautology.', 'start': 3524.954, 'duration': 1.902}, {'end': 3529.117, 'text': 'Finally, we have modus tollens.', 'start': 3527.636, 'duration': 1.481}, {'end': 3533.641, 'text': 'Modus tollens is P implies Q and not Q implies not P.', 'start': 3529.357, 'duration': 4.284}, {'end': 3535.838, 'text': "So let's break this down.", 'start': 3534.798, 'duration': 1.04}, {'end': 3537.639, 'text': 'P implies Q.', 'start': 3535.958, 'duration': 1.681}, {'end': 3543.38, 'text': 'P implies Q is only ever negative when the hypothesis predicts true and the conclusion is false.', 'start': 3537.639, 'duration': 5.741}], 'summary': 'Logical operations yield a true column, law of contradiction as tautology, and modus tollens explained.', 'duration': 50.83, 'max_score': 3492.55, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI3492550.jpg'}], 'start': 2368.039, 'title': 'Importance of logic in mathematics and propositions', 'summary': 'Discusses the significance of logic in mathematics, its role in deducing new information, and introduces composite propositions, logic algebraic laws, logic laws and statements, and tautologies with examples and analysis.', 'chapters': [{'end': 2500.516, 'start': 2368.039, 'title': 'Importance of logic in mathematics', 'summary': 'Discusses the importance of logic in mathematics and its role in deducing new information, parsing the meaning of sentences, and its significance for programmers and algorithms.', 'duration': 132.477, 'highlights': ['Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meaning of sentences, which is crucial for maintaining certainty and foundation in mathematics.', 'Programmers use algorithms, which are essentially just math, and mathematics requires logic, emphasizing the practical relevance of logic in the field of programming.', 'Introducing the notion of a proposition, defined as a declarative statement with a verifiable truth value, which serves as a fundamental concept for logical reasoning.']}, {'end': 2751.506, 'start': 2500.516, 'title': 'Introduction to composite propositions', 'summary': 'Introduces the concept of composite propositions, including conjunctions and disjunctions, and explains how to visualize their truth values using truth tables, with examples and analysis of verifiable truth values.', 'duration': 250.99, 'highlights': ['Composite propositions are declarative statements with verifiable truth values, made up of sub-propositions, including conjunctions and disjunctions.', 'Truth tables are used to visualize the truth values of composite propositions, with primitive propositions having two possible values, and conjunctions and disjunctions requiring additional rows to represent all possibilities.', 'Example analysis of composite propositions using truth tables, such as the conjunction P and Q and the disjunction P or Q, with explanations of their truth values based on the sub-propositions.']}, {'end': 3092.581, 'start': 2752.327, 'title': 'Introduction to logic algebraic laws', 'summary': 'Introduces algebraic laws of logic including idempotence, identities, complements, involution, and commutative law, demonstrating their logical equivalences through truth tables and definitions.', 'duration': 340.254, 'highlights': ['The chapter introduces algebraic laws of logic including idempotence, identities, complements, involution, and commutative law.', 'Demonstrates logical equivalences through truth tables and definitions.', 'The law of complements and the law of involution are explained using truth tables and definitions.']}, {'end': 3410.2, 'start': 3097.711, 'title': 'Introduction to logic laws and statements', 'summary': "Introduces the associative law, distributive law, de morgan's law, conditional statements, and universal and existential quantifiers in logic, expanding from two to three propositions, resulting in eight possible outcomes, and explaining the concepts of truth tables, logical equivalences, and quantifiers.", 'duration': 312.489, 'highlights': ["The chapter introduces the associative law, distributive law, De Morgan's law, conditional statements, and universal and existential quantifiers in logic, expanding from two to three propositions, resulting in eight possible outcomes.", 'The concepts of truth tables, logical equivalences, and quantifiers are explained.', 'Conditional statements are discussed, including their components and the evaluation of true and false outcomes.']}, {'end': 3607.261, 'start': 3410.2, 'title': 'Tautologies and false statements', 'summary': 'Discusses the concept of tautologies and false statements in mathematical propositions, providing examples and explanations, with a focus on the law of excluded middle, law of contradiction, and modus tollens.', 'duration': 197.061, 'highlights': ['The law of excluded middle states that p or not p is always true, demonstrated through a truth table with all combinations yielding true.', 'The law of contradiction is a tautology, as shown by the truth table for P and not P resulting in a column of truths after applying negation to the false column.', 'Modus tollens is illustrated through a truth table, demonstrating that it is a tautology due to its resulting column of truths and its nature of always yielding true statements.']}], 'duration': 1239.222, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/2SpuBqvNjHI/pics/2SpuBqvNjHI2368039.jpg', 'highlights': ['Logic is crucial for maintaining certainty and foundation in mathematics', 'Programmers use algorithms, which essentially rely on mathematics and logic', 'Introducing the notion of a proposition as a fundamental concept for logical reasoning', 'Demonstrates logical equivalences through truth tables and definitions', 'The chapter introduces algebraic laws of logic including idempotence, identities, complements, involution, and commutative law', "The chapter introduces the associative law, distributive law, De Morgan's law, conditional statements, and quantifiers in logic", 'The law of excluded middle states that p or not p is always true', 'Modus tollens is illustrated through a truth table, demonstrating its nature of always yielding true statements', 'Example analysis of composite propositions using truth tables, such as conjunctions and disjunctions', 'The law of contradiction is a tautology, as shown by the truth table for P and not P resulting in a column of truths']}], 'highlights': ['Discrete mathematics practical application in computer graphics', 'The importance of discrete mathematics in computer science and programming', 'The chapter introduces the concept of sets and set notation', 'The chapter discusses interval notation and its efficiency', 'The chapter demonstrates a proof that the square root of 2 is irrational', 'The chapter introduces four binary operators for sets', 'Logic is crucial for maintaining certainty and foundation in mathematics', 'The law of complements states that if we take the union of a set and its complement, it equals the universe', 'The distributive law states A intersection B union C equals A intersection B union A intersection C', 'The chapter introduces algebraic laws of logic including idempotence, identities, complements, involution, and commutative law']}