Full transcript of “Everything You Need To Know About Set Theory | All-in-One Video”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:00 A set is simply a collection of distinct objects that we group together and think of as a single whole. These objects are called elements or members of the set. The key word here is distinct which means no element repeats inside a set and order does not matter. For example, if I say the set of colors on a traffic light, we can have one set A whose collection is red, yellow, green and another set B whose collection is green, red, yellow, red. So both the sets are equal because order does not matter and repeated elements are ignored in a set. A set is well defined meaning that for any given object we can clearly decide whether it belongs to the set or not. For example, the set of months of the year is well defined because we can always say whether a name like April is in it. While something vague like the set of beautiful flowers is not well defined because different people may disagree on which flowers count as beautiful. In fact, the elements of a
1:05 set do not need to be related at all. You can easily have a set containing one person, two flowers, a laptop, and the number 10. And it is still a proper set because we can clearly say what is in it. The only condition is that the collection must be well defined. So saying the set of one person and two flowers is perfectly fine. But calling it the set of nice things is not since the word nice is subjective and not precise. A set can be represented in mainly two ways. Roster form and set builder form. In roster form, also called tabular form, we simply list out all the elements of the set inside curly brackets separated by commas. For example, the set of vowels in English can be written as this. On the other hand, we have a set builder form where we do not list all the elements one by one, but instead describe them by a property or condition that they all satisfy. For example, the same set of vowels can be written as the set of all letters X
2:14 such that X is a vowel in English. When we say the set of all letters X, we are introducing a symbol here X that stands for any possible element we might consider. Then we add a condition which is such that X is a vowel in English. This condition acts like a filter. It tells us which letters are allowed inside the set. Here we are collecting every letter X from the English alphabet but we only keep those X that satisfy the rule of being a vowel. The power of this form is that we do not have to write out all the members when the set is very large or even infinite. For instance, the set of all whole numbers greater than five can be written as the set of all numbers X such that X is a whole number and X is greater than five. By the way, this symbol means such that. Then this symbol means belongs to or it shows X belongs to W which is whole number symbol and this represents and. So we say this as X such that X belongs to a whole number and X is greater than
3:24 5. Clearly in roster form we would have to write 6 7 8 9 and so on forever which is impossible. But in set builder form, a single clear rule captures the entire infinite collection. By the way, we can also represent such that using this vertical bar symbol instead of a colon. So both these set builder forms are correct. Now let me know in the comments how would you write in set builder form the collection of all even numbers greater than 10. Now when we start looking at sets more closely, we see that they can be of different types. A set is called finite if it contains a limited number of elements like the set of months in a year. While a set is infinite if the counting of its elements never ends like the set of whole numbers. Now we have already defined what we mean when we say that two sets are equal like they have exactly the same elements no matter the order or repetition. Two sets are said to be equivalent if they have the same number of elements even if the elements
4:35 themselves are different. For example, the set of vowels and the set of numbers 1 to five like this are equivalent since both contain five elements. Then we have some special type of sets like a singleton set which has exactly one element like the set of the number five alone while an empty set or null set has no elements at all like the set of months with more than 31 days. It will be an empty set. Right? Another important idea is that of subsets. If every element of a set A is also an element of a set B then we say A is a subset of B. For example, if A is the set 1 2 and B is the set 1 2 3 and four, then every element of A is also in B. So A is a subset of B. This is the symbol we use for subset. In the same way, if A is the set red, blue, and B is the set red, blue, green, and yellow, then A is a subset of B. Now an empty set is considered a subset of every possible set because there is no element in it that can ever violate the rule. Now if A
5:49 is a subset of B but B has at least one element not in A, then A is called a proper subset of B and we denote it like this. For example, if A is the set 1 2 and B is the set 1 2 3, then A is a proper subset of B because B has the extra element three that is not in A. Now, here's the difference between subset and proper subset. In subsets, A could be smaller than B or exactly equal to B. But in proper subsets, B must have at least one extra element. Now hey we can even have a set of sets meaning a set where each element itself is a set like a set whose members are the set of vowels and the set of consonants. This idea combined with the idea of subsets gives us the power set which is the set of all possible subsets of a given set. For example, if X is the set 1 2, then the power set of X is the collection of four subsets. The empty set, the set containing one, the set containing two, and the set containing one and two
7:06 together. All of these sets are subsets of set X. And this one set, which is the set of all such subsets, is a power set of X. Now, a superset is about comparison between two sets. If every element of set A is inside set B, then B is called a supererset of A. Example, B is 1 2 3. A is 1 2. Here B is a supererset of A. So a superset is simply the opposite of a subset. And a proper superset is the opposite of a proper subset. Finally, we also have the universal set which is a context dependent supererset of all sets. For example, if in a classroom of 50 students, if A is the set of students who play cricket and B is the set of students who play football, both A and B are subsets of the universal set U, which is the entire class of 50 students. Next, let us talk about operations on sets, which are simple ways to combine or compare different sets and they help us study how sets relate to each other. First operation is the union of two sets which gives us a
8:22 new set that contains every element which is in either of the sets or in both. For example, if A is 1 2 3 and B is 3 4 5 then their union is 1 2 3 4 and 5. As another example, if A is apple, mango and B is mango, banana, then their union is apple, mango and banana, then the intersection of two sets is the new set that contains only those elements which are common to both sets. For example, if a is 2 4 6 and b is 4 6 8 then their intersection is 4 6. Similarly, if A is cat, dog, and B is
9:09 cow, goat, sheep, then the intersection is an empty or a null set. The next operation is the difference of two sets, which is the set of elements that are in the first set but not in the second. For example, if a is 1 2 3 4 and b is 34 5, then a minus b is 1 2 because 1 and two are in a but not in b while three and four are not included since they are also in b. On the other hand, B minus A is five because five is in B but not in A and the other elements of B are excluded since they are also in A. As another example, if A is red, blue, green and B is blue, then A minus B is red, green. Now let me know in the comments what set do we get after B minus A? Then the complement of a set is the set of all elements in the universal set that are not in the given set. For example, if the universal set U is 1 2 3 4 5 and A is 2 3, then the complement of A is 1 4 5. As another example, if the universal set is mango, apple, banana,
10:29 orange, and B is apple, banana, then the complement of B is mango, orange. We have the algebra of sets, which is a collection of simple laws that sets follow very similar to rules in arithmetic. The commutative law says that A union B is the same as B union A and A intersection B is the same as B intersection A. For example, if A is 1 2 and B is 2 3 then both A union B and B union A give 1 2 3 and both A intersection B and B intersection A give two. The associative law says grouping does not matter. So A union B union C is the same as A union B union C. And similarly for intersection for example if A is 1, B is 1 2 and C is 2 3 then B union C is 1 2 3 and its union with A is again 1 2 3. Now for this part A union B is 1 2 and its union with C is 1 2 3. So hey both of them are the same. And if A is 1 2 B is 2 3 and C is 2 then both A intersection B intersection C and A intersection B intersection C give two. The distributive law explains how union
12:06 and intersection mix. So A intersection B union C equals A intersection B union A intersection C. For example, if A is 1 2 B is 2 3 and C is 3 4. First let us check the left hand side. This B union C will be 2 3 and 4 and its intersection with A will be 2 only. Now for the right side A intersection B is 2 and A intersection C is a null or an empty set. So union of both of them will be two itself and that's it. And also A union B intersection C equals A union B
12:52 intersection A union C. For example, if a is 1, b is 1, 2 and c is 1, three both sides reduce to one. Then we also have deorggan's identities that relate complements with unions and intersections. Namely, the complement of a union b equals the intersection of the complement of a and the complement of b and the complement of a intersection b equals the union of the complement of a and the complement of b. For example, in the universal set U equal to 1 2 3 4. If A is 1 3 and B is 3 4. First we find the union of A and B which is the set of all elements that are in A or in B or in both. So A union B is 1 3 4 because 1 and three come from A and three and four come from B combining all without repetition. Now the complement of A union B is the set of all elements in the universal set that are not in A union B. Therefore the complement of A union B is just two. Next we calculate the complements of A and B individually. The complement of A
14:11 is all elements in the universal set that are not in A. Since a is 1 3 removing these from u gives a complement equal to 2 4. Similarly the complement of b is equal to 1 2. Finally we find the intersection of a complement and b complement which is the set of elements common to both complements which is two because two is the only element that appears in both sets. Hence verified this equation. Now please do the same for this one. Hey folks, do not fear if you see expressions with too many operations. Just break the expression down step by step. Work inside the brackets first. Evaluate complements and differences next, then intersections and finally unions. Now let us talk about a vin diagram. It usually has a rectangle that represents the universal set and inside it we draw circles to represent the sets themselves and shaded regions or overlaps make the operations on sets very easy to understand. For the two set cases, we draw two circles inside the rectangle.
15:27 For example, if one circle A represents students who play cricket and the other circle B represents students who play football, then the overlapping region represents students who play both cricket and football. The full area of both circles together represents students who play at least one of the two games. Meaning the student either plays football or cricket or both. And the part of a alone represents students who play only cricket but not football. So tell me in the comments what this part of B represents. Now the complement of A represents people in the universal set who do not like cricket at all or this shaded region. Okay, just for fun let us see Deorgan's union complement law using vin diagram. See this is a union B. So its complement will be this region. Now the complement of A is this and also the complement of B will be this. So the intersection of both of them will be this region which verifies this law. Wow, nice. For the three set
16:36 case, we draw three overlapping circles inside the rectangle. The overlapping regions become more detailed. For example, if one circle A represents students taking maths, another circle B represents students taking science and the third circle C represents students taking English. Then the central overlap of all three circles represents students taking all three subjects together and each pair-wise overlap like A intersection B represents students taking both maths and science but not necessarily English. and the entire union represents students who are taking at least one of the three subjects. Then this individual part of A represents a student who is only taking maths and nothing else. Also, the area outside all circles within the rectangle represents people in the universal set who have not taken any of these three subjects. Now, vin diagrams are especially useful in solving word problems where numbers are given about groups and their overlaps.
17:44 For example, suppose in a class of 50 students, 30 like cricket, 25 like football, and 10 like both cricket and football. If we want to know how many students like either cricket or football or both, we can use a ven diagram. Draw a rectangle for the class of 50 students. Then draw one circle A for cricket and another circle B for football. Place 10 in the overlapping region because these are the students who like both. Now since 30 like cricket in total, the remaining 20 must be in the cricket only part of circle A. Similarly, since 25 like football in total, the remaining 15 must be in the football only part of circle B. So the total number of students in A union B is 20 + 10 + 15 which is 45. This means 45 students like cricket or football or both. And the remaining five students in the rectangle but outside both circles represent those who do not like either game. Also the complement of A means students who do not like cricket. Out of 50 students, 30 like
19:01 cricket. So the other 20 do not like cricket. These 20 are made up of 15 who like only football plus five who like neither. So A compliment has 20 students. Let me know in the comments how many students will be there in the complement of B. Finally, before we end this video, we will talk about the disjoint sets and cardality of sets. Two sets are called disjoint if they have no elements in common, meaning their intersection is empty. The cardality of a set written as n of a simply means the number of elements in the set a. For example, if a is a set of all vowels, then its cardality is five because there are five vowels. In the case of infinite sets, we say the cardality is infinite. For instance, the set of all natural numbers has infinite cardality. I am super tired right now. So, let's pause here. But if this video gets 5,000 likes, then in the next video I will take you further into the exciting journey of set theory where we move from operations on sets into functions and
20:18 mappings exploring what functions really are, their domain, co-domain and range and the different types like injective, sjective and bjective along with composition and inverses. And after that we will dive into some advanced concepts of set theory like infinite sets and countability comparing countable sets such as natural numbers integers and ration with uncountable sets such as the real numbers and even touching upon deeper ideas like Russell's paradox. So good. [Music]
