Mark each of the following true or false.
___ a. If * is any binary operation on any set S, then a * a = a for all a ∈ S.
___ b. If * is any commutative binary operation on any set S, then a * (b * c) = (b * c) * a for all a, b, C ∈ S.
___ c. If * is any associative binary operation on any set S, then a* (b * c) = (b * c) * a for all a, b, c ∈ S.
___ d. The only binary operations of any importance are those defined on sets of numbers.
___ e. A binary operation * on a set S is commutative if there exist a, b ∈ S such that a * b = b * a.
___ f. Every binary operation defined on a set having exactly one element is both commutative and associative.
___ g. A binary operation on a set S assigns at least one element of S to each ordered pair of elements of S.
h. A binary operation on a set S assigns at most one element of S to each ordered pair of elements of
S.
___ i. A binary operation on a set S assigns exactly one element of S to each ordered pair of elements
of S.
___ j. A binary operation on a set S may assign more than one element of S to some ordered pair of elements of S.