State whether the following statement is true or false and justify your answer: If $^*$ is a commutative binary operation on $N$,then $a ^* (b ^* c) = (c ^* b) ^* a$.
(A) Given that $^*$ is a commutative binary operation on $N$,which means $x ^* y = y ^* x$ for all $x, y \in N$.
Consider the right-hand side $(RHS)$:
$RHS = (c ^* b) ^* a$
Since $^*$ is commutative,we can write $(c ^* b) = (b ^* c)$.
So,$RHS = (b ^* c) ^* a$.
Now,by the commutative property,we can swap the elements around the operation $^*$:
$(b ^* c) ^* a = a ^* (b ^* c)$.
Thus,$RHS = a ^* (b ^* c) = LHS$.
Therefore,the statement is true.
Consider the right-hand side $(RHS)$:
$RHS = (c ^* b) ^* a$
Since $^*$ is commutative,we can write $(c ^* b) = (b ^* c)$.
So,$RHS = (b ^* c) ^* a$.
Now,by the commutative property,we can swap the elements around the operation $^*$:
$(b ^* c) ^* a = a ^* (b ^* c)$.
Thus,$RHS = a ^* (b ^* c) = LHS$.
Therefore,the statement is true.
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