Question

Which one below is true?

• A. Every binary operation, $*$ on a set satisfies the identity axiom.
• B. Every binary operation, $*$ on a set satisfies $a *(b * c)=(a * b) * c$.
• C. Every commutative binary operation is associative.
• D. Every binary operation defined on a set having exactly one element is both commutative and associative.

Answer 1

Answer: (D)

Addition' is a binary operation on $N$.
Under $+, N$ has no identity element
Under this operation, $G$ is an abelian group. 'subtraction' is a binary operation on the set $Z$. But $a-(b-c) \neq(a-b)-c$.
The operation $*$ defined on $N$ by $a * b = a ^{ b }+ b ^{ a }$ is both binary and commutative. But this is not associative (verify this ).