About the definition of binary operation on a set, in my notes it says a binary operation on S is a map $*:S\times S\to S$, it does not have to be a function, it is a mapping. But in the textbook, it says while defining a binary operation on a set $S$, we must be sure that exactly one element is assigned to each possible ordered pair of elements. Doesn't that mean it is a function? I am confused here, does a binary operation have to be a function? Thanks
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