Determine whether the description of is a valid definition

Determine whether the description of # is a valid definition of a binary operation on the set:

[a] On R, where a # b is a * b (ordinary multiplication).

[b] On Z, where a # b is a^b.

Solution

A binary operation is a rule for combining two objects of a given type, to obtain another object of that type.

[a] If a and b are two arbitrary elements of R , then a # b is a*b ( ordinary multiplication), then a # b = a*b R. Thus, the description of # is a valid definition of a binary operation on R .

[b] Z is the set of integers. If a and b are two arbitrary elements of Z, then a # b = a^b also belongs to Z . Thus, the description of # is a valid definition of a binary operation on Z.