Question:
For $a l l a, b \in N$, we define $a * b=a^{3}+b^{3}$.
Show that $*$ is commutative but not associative.
Solution:
let $a=1, b=2 \in N$
$a^{*} b=1^{3}+2^{3}=9$
And $b * a=2^{3}+1^{3}=9$
Hence * is commutative.
Let c = 3
$\left(a^{*} b\right)^{*} c=9^{*} c=9^{3}+3^{3}$
$a^{*}\left(b^{*} c\right)=a^{*}\left(2^{3}+3^{3}\right)=1 * 35=1^{3}+35^{3}$
$\left(a^{*} b\right)^{*} c \neq a^{*}\left(b^{*} c\right)$
Hence * is not associative.