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A mathematical operation, such as addition or multiplication, performed on two elements of a set to derive a third element
So, the basic requirements are this: There must a be a set stated, we can call it S. There must be an operation, we can call it +, that is a rule for taking two elements and deriving a third element. For any two elements, a and b in S, a+b must be an element of S.
And actually, if you get into abstract algebra in depth at all, they tend to use star more often than plus or x or any other symbol.
But basically here is the jist of it. To name a binary operation, give a set and an operation. The condition that will disqualify something as a binary operation is if there are two things that you could perform the operation on and get something outside of the set.
Example: The set is the positive integers. The operation is normal addition. Is this a binary operation? Well, if I choose any two things from the set and I add them, the result will be a positive integer, so yes. It's a binary operation. Example: The set is the positive integers. The operation is normal subtraction. Is this a binary operation? Well, consider the operation (3-8). 3 and 8 are both in the set, but the result is -5, which is outside of the set, so this is not a binary operation.
ok there is any website or a video which can tell me more about this...
Im not quite sure we would not call that a binary operation. But we would say that it is not closed on that set. In other words, Im not sure that the set must be cloased for it to be considered a binary operator. This would be the CLOSED operation on S
i didn't understanad what you want to say @zzr0ck3r
Zzrocker, closure actually is a requirement of the definition. http://www.math.vt.edu/people/kohler/Sect5_4.pdf
great that helps:)