Hello friends. In this video, we will learn what binary operations are, some examples of binary operations, and what commutative and associative binary operations are. First of all, what binary operations are. A binary operation star on a non-empty set X is a mapping star from X cross X to X, that is a binary operation on a set X is a mapping whose input is an ordered pair of set X and output is also an element of the set X. What does it mean? Suppose you have set of real numbers. We all know what real numbers are. Suppose you define operation star as addition of real numbers. If you take two real numbers, say a and b, you add them. Addition of these two real numbers is again a real number. We can say that this operation plus all the real numbers is a binary operation. Because here, what I am taking X here, I am taking X as R. This X is basically R here. I'm applying this operation is star, which is plus in this case on R cross R to R. That means I am taking two elements, a and b of R, applying this operation plus on these real numbers, and the resultant is also a real number. Hence, I'm saying that this plus is a binary operation. Now we take another example. Usual additional set of integers is also a binary operation. We already know what integers are. Integers are basically plus minus 2, plus minus 1, 0, and so on. These are all integers. If you take two integers, say x and y, and you add them, usual addition, then addition of two integers is also an integer. We can say that this is a binary operation. If you define operation star as a plus b by 2, where a and b are from the set X and X is the set of natural numbers. If you take two natural numbers like 2 and 3, for example, this is star. Then this star is defined as 2 plus 3 upon 2, which is what? Which is 5 by 2. This 5 by 2 is not a natural number. We can say that this operation star or this definition is not a binary operation. Because it is not closed. A binary operation star on a non-empty set X is said to be commutative if x star y equal to y star x for every x, y in X. If this property holds for every x, y in X, then we say that that property is the commutative property. Associative property, is operation star is said to be associative if x star y star z equal to x star y star z for every x, y, z into X. What does it mean? It means that you first operate that defined operation on these two elements, y and z, and whatever you will get, you apply that operation on x to get the left-hand side. Similarly, first of all, you apply that operation on x and y. Whatever result you obtained from here, you apply that with z to get the entire expression in this. If these two are equal, then we say that it satisfy associative property. Here are some examples. Suppose you take usual additional set of real numbers. We say that it satisfies commutative and associative property. You take two real number, say 2 and 3. You take 2 plus 3. You take 2 plus 3 or 3 plus 2, both are same. In general, if you say x and y are any two real numbers, then x plus y is always equal to y plus x for any x and y. We say that this satisfy commutative property. For associative, if you take three real numbers, x, y, and z. If you take x plus y plus z, you take bracket here, or you take bracket here, x plus y plus z. Both are equal. You add first of all, x and y. Whatever you obtained here you apply you add with z this value. First of all, add y and z. Whatever you obtained from here, you add with x, this value. If these two are equal, for any three real numbers. We say that real numbers satisfy commutative and associative property for the binary operation in usual addition. Similarly, we can show that multiplication of real numbers also satisfy commutative and associative property because we know that x into y is same as yx for all x, y belongs to real numbers. Similarly, this is commutative. Similarly, x into y into z is same as x into y into z for all x, y, z in R. On the other hand, if you see usual subtraction on the set of integers, it does not satisfy commutative as well as associative properties. Why? You can see one counter example. For example, if you see 1 minus 2. It is not equal to 2 minus 1. That is x star y is not equal to y star x. It is minus 1, it is 1. This means this operation, minus, it is star here. This operation does not satisfy commutative property. Similarly, for associative, if you take, say 1 minus 2 minus 3 or you take 1 minus 2 minus 3. First of all, solve the bracket. This bracket is minus 1. This is 1 plus 1, this is 2. Left hand side is 2. You solve right hand side, it is minus 1 minus 3, this is minus 4. These two value are not same. These two are not same, so we see that it does not satisfy associative property. There are some other examples also. Suppose you take set of all two cross two real matrices, and you define the operation star as, a star b equal to a plus b for every A, B in M, then this star satisfy associative as well as commutative property. How we can say this? We know that if you take two matrices A and B, usual addition if you perform, then A plus B is same as B plus A. It holds for every A, B in M. That means it satisfy commutative property. Similarly, if you take three matrices, A, B and C, and if you perform this operation, A plus B plus C, or you take A plus B plus C like this for every A, B, C in M, then also this equality holds. We say that this operation, which is usual addition of matrices satisfy associative as well as commutative property. On the other hand, if you say usual multiplication of matrices, say A into B. If you see this binary operation, then this satisfy associative property, but it doesn't satisfy commutative property. We have examples. If you take two matrices, say A has 1,2,2,3 or B has some other matrices say, 2 minus 1,3,1, then we can verify very easily that AB is not equal to BA. It does not satisfy commutative property. But if you see A into B, C is equal to A, B, C. This we can easily verify for any two cross two matrices. In this way, we can see this, that this operation satisfies associative property but does not satisfy commutative property. In this video, we have learned what binary operations are. We have also seen some examples which do not satisfy binary operations. Then we have seen some properties of binary operation like commutative and associative.
