Binary operation (Mathematics in the Modern World, GE Curriculum Higher Education)

Binary operation. These two words, they are not entirely new to you, but put together, it's the first time that you see them put together. So binary has to do with things that are composed of two things. So the prefix there, it's what you call it, prefix. The prefix by indicates something that is two. The operation here has to do or somewhat comparable with your understanding of the basic mathematical operations. So what are they? They could be something like addition, subtraction, multiplication, or division. So what is a binary operation? Let G be a non-empty set. The binary operation star is a well-defined rule for a pair of elements A, B in G such that A star B is element of G. And you know, in other books, this is how they write a binary operation using mathematical symbols. G cross G, you have an element of G. So that is how you read it. You don't read it as G times G. G cross G, what it means is you are pairing elements of G with the other elements that is also in G. And that is what happens here. So you will pair the two elements and then it will produce a result. The outcome of that pairing depends on how you define your operation star. And the result is also to be found in set G. This is maybe the first time you are seeing a sentence that is constructed in this way. But I'm telling you, this is something that you had been doing, we had been doing ever since we stepped into school. But the operations here, the binary operations that you were doing were the operations of this one. Addition, subtraction, multiplication and division. But we didn't call them binary operations. We just call them mathematical operations. And this set and the set that you were working on was the set of real numbers. And our symbol for that is R. The binary operation thing is not entirely new. Maybe the titling, you know, how it's called is new. But we had been doing this ever since we went to school. Let G be the set of integers. Addition modulo n. Now that you have graduated from high school, it's about time to be introduced to new operations. When you were in high school, the basic arithmetic were the only operations that you knew. But now, you know this. Addition modulo n. So it's not exactly like the usual addition that we know. So what we do with addition modulo n arithmetic is we divide, we divide the sum by 2 and we look for the remainder. So in this case, 5 plus 8 addition modulo 2 is equal to 1. Why? Because when we divide 13 by 2, the remainder is 1. 10 plus 22 modulo 2 is equal to 0 because it is an even number. The sum is an even integer. And when you divide it by 2, the remainder is 0. 5 plus 8 modulo 7 is equal to 6. 5 plus 8 is 13. You divide it by 7, the remainder is 6. So you see, we have a rule on how to do the operation, the operation of addition modulo n. We can proceed with multiplication and the idea is the same. When we apply the operation of multiplication modulo 2 or modulo n, again, we get the product but we divide it by n. In this case, by 2 and we look for the remainder. So this one is obviously equal to 0. 11 times 22 multiplication modulo 5. So 11 times 22 is 242. You divide it by 5, okay? So we don't have to proceed with the entire operation. We know that the remainder is going to be 2. This is going to be your r. 5 times 8 multiplication modulo 7 is 5. But there are other more operations. For example, operation mean denoted by this symbol. So it's like an arrow that is pointed downwards. And so this is how we define this operation. A min b is equal to the minimum of the two integers. So 5 min 8 is equal to 5. 11 min 22 is equal to 11. Negative 11 min negative 120 is equal to negative 120. So I hope that you are noticing that when we when we do a binary operation, first, of course, we have to give it a name. Look at this. We are giving it a name and a symbol and then we are defining it because it is a rule. We must lay down the rule about how to do the operation. And one more thing, okay? I am forgetting one important criterion of a binary operation is the closure property. When you apply the operation on those integers, the result is also to be found in the set of integers. So if we have operation mean, we can also produce an operation called max. We can use this symbol to denote max. A max b is equal to the maximum. It is equal to the larger integer of the two integers. Let g be the set of integers. We shall define o plus. Okay, not plus because we already know what what plus is supposed to do. But this one is not an ordinary addition and this is our symbol. So this is how we shall define o plus. A o plus b is equal to this. 8 times b plus a plus b. So what is 5 o plus a? Well, you go back to this. Go back to this and you use the rule. So that's equal to 5 times a plus 5 plus 8. What is 5 o plus negative 18? Again, go back to the rule. Apply the rule. This is equal to 5 times negative 18 plus 5 plus negative 18 which is equal to negative 103. o slash. So let o slash be our operation and our elements are taken from the set of integers. And this is how we are defining o plus. o plus b. No, a o plus b is equal to a plus b times a minus b. So what is 5 o plus 7? What is 5 o plus 7? So this one is going to be 5 plus 7 times 5 minus 7. And this is equal to negative 24. This is 12. This is negative 2. So 12 times negative 2 is negative 24. How about this? 10 o plus 17. So again, you go back to the rule. You apply the rule on these integers. So it's 10 plus 17 times 10 minus 17. So this is 27 times negative 7. And this is equal to negative 189. So that's the idea of the binary operation. So we have been doing binary operation ever since we stepped into school, grade school. But you didn't know it was called binary operation.