Help for Abstract Algebra?

Question

True or False?

1.)A binary operation on a set S assigns at least one element of S to each ordered pair of elements of S.
2.) A binary operation on a set S assigns at most one element of S to each ordered pair of elements of S.
3.)A binary operation on a set S may assign more than one element of S to each ordered pair of elements of S.

Can someone tell me why it makes the statement T or F? Thanks!

Answer 1

1∧2∧¬3

Remember, a binary operation * on a set S is simply a function *:S×S → S. A function associates each element of the domain with exactly one element of the codomain. Exactly one means "at least one and not more than one." So 1 and 2 are true, and 3 is false.