Question

For each binary operation $\ast$ defined below, determine whether $\ast$ is binary, communtative or associative.
(v) On $Z^{+}$, define $a\ast b=a^b$

Answer 1

On $Z^{+},\ast$ is defined by $a\ast b=a^b,$
$a^b\in Z^{+}$, so the operation $\ast$ is binary operation.
It can be observed that $1\ast 2=1^2=1$ and $2\ast 1=2^1=2$
Therefore, $1\ast 2\ne 2\ast 1$ where $1,2,\in Z^{+}$
Therefore, the operation $\ast$ is not commutative.

It can also be observed that
$(2\ast 3)\ast 4=2^3\ast 4=8\ast 4=8^4=(2^3{)}^4=2^{12}2\ast (3\ast 4)=2\ast 3^4=2\ast 81=2^{81}$
Therefore, $(2\ast 3)\ast 4\ne 2\ast (3\ast 4);where2,3,4\in Z^{+}$
Therefore, the operation $\ast$ is not associative.