Transcription
[Music] So, let's begin our topic for today: that is, set theory. We begin our topic for today, that is, set theory. So, let's begin with absolutely the basics. What is a set? The definition of a set: a set is a collection of distinct objects or elements. It has to be distinct; it can't be the same. Okay. An example would be: set A is the letters a, b, and c. Okay, this is a set.
There are a couple of set formats in which you can express a set: set builder format, roster format. Okay. Now, what is a set builder format? A set builder format is describing the elements of a set using a defining property. Describing the elements of a set using a defining property. What does this mean? Set A is all X such that X is between 5 and 8, where X is a natural number. Okay. X is such that X is lying between 5 and 8, where X is a natural number. In the roster format, the same thing will be written as 5, 6, 7, 8. Both of these are describing exactly the same thing, but one is in a set builder format, the other is a roster format. The set builder format is usually used in order to express an infinite set, like, for example, set A is the set of all natural numbers. Okay. Okay. In the roster format, you'll have to write it as 1, 2, 3, 4… which is not very self-explanatory. Okay. So, this is basically the difference between the set builder format and the roster form.
So, we have understood what is the definition of a set. We have understood the different formats in which we can describe a set. Now is the chance of going into the different kind of sets that are possible. The different kinds of sets possible are as follows: empty set, singleton, finite, infinite. Okay. So, these are a few different types of sets that are possible. Empty set. What is an empty set? An empty set is a set which does not have anything in it. In the roster format, we write it as: it's a null set. Okay. You can write it in this way, or you can put it as an example: how can X be less than 4 and greater than 8 at the same time? Not possible. So, it's an empty set. Okay. So, that's an empty set. Singleton set. Okay. Now, what's a singleton set? A singleton set is a set that has exactly one element in it. Okay, only one element in it. An example: only the number two. Finite sets are those sets that have a finite number of elements in it. Okay. Like, for example, the set of natural numbers is infinite; the set of real numbers are infinite. Finite is the example that we have. A singleton set is a finite set. A finite number of elements are there: 5, 6, 7, 8. This is an example of a finite set. This is an example of an infinite set. Okay. Finite set can be 5, 6, 7, 8. Infinite set belongs to natural numbers. Okay. These are all examples of different kinds of sets, right? Two more types of sets that I wanted to share were equal sets. What's an equal set? An equal set is a set that are equal, as simple as that. Like the set 1, 2, 3 is the same as the set 3, 2, 1. Even if the order is different, it doesn't matter; the set is the same. A and B are equal sets. Equivalent sets. Equivalent sets are the sets where the number of elements are equal. So, if A and B both have three elements, let A be 1, 2, 3; let B be 4, 5, 6. A and B are equivalent. The number of elements are the same; that's all. It does not matter that the elements in itself are they the same or not; that does not matter. Okay. The number of elements are the same; the element values may be different. That's an equivalent set. Yes. 1, 2, 3 will be the same as 1, 2, 3, 3, if I put three two times. Okay. Like A and B are the same, and C is also the same, like 1, 1, 2, 2, 3. It's also equal to C. I'm repeating one. A set is a collection of distinct objects or elements; it has to be distinct. One and one, one is the same. So, C is the same as 1, 2, 3. So, that is an equal set, right? That was the type of sets. Now, let's go into categories of sets. These were the different types of sets. Now, let's go into categories of sets. Categories number one category: Universal set. Universal set is the set that contains all elements under consideration. Okay. It contains all elements under consideration. So, an example of a universal set will be: if I consider a set of all numbers, set of all numbers, set of all real numbers. Okay, that is a universal set. Set of all vegetables: universal set. Okay. That's a category of set. Category: subset. Subset is a set that contains… subset is a set that contains few elements from another set. Like, for example, there are two sets, A and B. A has, say, 1, 2, 3; B has, say, only the element two. So, B is called a subset of A. Okay. And in subset also there are two different categories: one is proper subset, the other is improper subset. Proper, improper. Proper is: if at least one element in the subset… at least one element in A is not in the subset. Okay. So, this is an example of a proper subset. B has only the element two; A has 1, 2, 3. So, A has at least one element that is not there in the subset. So, A having at least one extra element than the subset will be called a proper subset. An improper subset will be: if B would contain 1, 2, 3. Basically, A and B are the same. Okay. So, if B, B is equal to A, if the set B and A are equal, then B is an improper subset of A, and A is an improper subset of B. Whenever A and B are equal, they are called improper subsets only. Okay. Universal set, subset. Subset concept of superset. Okay. The concept of superset. Superset, sir. Superset. Superset is: A is 1, 2, 3; B is two. So, A is called the superset of B; that is, A contains all the elements in B and more. A contains all the elements in B and more. So, A is called the superset of B. Universal, subset, superset concepts. Concepts: a universal set, a subset, or A has a superset. What are the signs of proper subset and subset? Subset, subset may categories: category proper or improper. Proper: that B is called a proper subset of A if A will contain one extra element than B, at least one extra element. Okay. That is when… when you get a proper subset, and improper is when A and B are both equal. Okay. So, this is what we see when we talk about categories of sets: different categories of sets. Next into operations of sets. Let's now talk about operations. Definition: set, different forms: set builder format and the roster format. We saw the different kinds of sets that are possible, and we saw the different categories of sets that are possible. Now, we go into the next aspect, that is, operations of sets. Operations, operation: Union. A is 1, 2, 3; B is 4, 5. A union B, this is how you give a notation, is 1, 2, 3, 4, 5. Un-combining elements from two or more sets into a single set. Elements comb… com elements will be there in the Union. Okay. And obviously, a set is a collection of distinct objects, right? Two same objects… so even if you have A as 1, 2, 3; B is, say, 3, 4, 5; A: 1, 2, 3; B: 3, 4, 5; to A union B will be 1, 2, 3, 4, 5. Okay. Second operation that I want to talk about is intersection. Okay. Same example: A is 1, 2, 3; B is 3, 4, 5. A intersection B is the element three. That I will consider only those elements that are common to the sets under consideration. So, finding all the elements that are common. So, A and B have one element in common, that is the number three. So, that is what the intersection… section will… will be giving me. Union: combining elements from two or more sets. Intersection: finding elements common to two or more sets. So, Union is combining elements; intersection is finding the common elements. Okay. Third operation: difference. Are we… difference. A is 1, 2, 3; B is 3, 4, 5. A difference B, also written as A – B: all the elements in A but not in B. So, A – B will be one, two only. So, what is difference? Finding elements in one set but not in another. Finding the elements in one set but not in another. That is what I mean by difference. So, elements… so I will get one, two only. Similarly, B – A will be giving me what? Can you tell me? Put it in the chat box. Think about the definition: finding elements in one set but not in another. Finding the elements in one set but not in another. It is all the elements in B but not in A. So, the elements are four and five only. So, the elements that are in B but not in A. Three here, three is the common element between A and B. Okay. That’s the third operation that I want to talk about, which is difference. Fourth operation that I want to talk about in the operation of sets is complement. Complement: finding… finding elements not in a given set within the universal set. Finding those elements which are not there in a given set but they are in the universal set. So, if I consider the universal set as 1, 2, 3, 4, 5, 6, and I consider a set A that is 1, 2, 3, A complement will be basically 4, 5, 6: those elements that are not in A but in the universal set. Universal set. Universal set is, say, 1, 2, 3, 4, 5, 6. A is 1, 2, 3. A complement are all the elements that are in the universal set but not in A. Clear? Clear. Then please raise your hand. That is the universal set. Operations: Union, intersection, difference, and complement. Okay. Can you also tell me if I have another set B, say the set B is 3, 4, 5, 6; that is the set B. So, A union B is the universal set. Okay. A union B is the universal set. A – B… Union, intersection, and complement. Can you tell me what will A – B be in the notation of either A union or intersection or complement? Similarly, please tell me what will B – A be? I want to eliminate the minus part. A – B can also be written as A – A intersection B. That’s correct, but I don’t want to write minus. I want to express A – B as a combination of only Union, intersection, or complement. I don’t want to put a negative sign at all. So, what is the notation for A – B? First question. Second question: what is the notation for B – A? A – B can be represented as A intersection B complement. B complement and A intersection with that. That means that everything in the universal set but not in B and common with A; that is A – B. And B – A… no prizes for guessing: B intersection A complement. So, this is how you express the difference with intersection, Union, and complement. Okay. There are various op… operations on the operation of sets also. Like, for example, A union B whole complement can be written as A complement intersection B complement. Can anyone tell me who first basically derived this, and it’s named after him? It’s called… law. Can you specify in the chat box? It’s what law? It’s De Morgan’s law. That’s correct. It’s De Morgan’s law. Okay. A union B complement is A complement intersection B complement, and A intersection B complement is A complement Union B complement. Okay. You can see this with this example also. A union B whole complement here will be the null set. So, A complement intersection B complement will be null. A intersection B whole complement. A and B have only one thing in common: three. So, their complement will be 1, 2, 4, 5, 6. A complement is 4, 5, 6. B complement is 1, 2, 4, 5, 6. A complement Union B complement. Okay. So, both of these are De Morgan’s law: the relationship between intersection, Union, and complement. So, that’s about the operations of sets. [Music]