State whether the following statement is true or false. Justify: For an arbitrary binary operation $^*$ on a set $N$,$a \,^* \,a = a$ for all $a \in N$.

(B) The statement is false.
To justify this,consider a counterexample.
Let the binary operation $^*$ on the set of natural numbers $N$ be defined as $a \,^* \,b = a + b$ for all $a, b \in N$.
Now,consider an element $a = 3 \in N$.
According to the given condition,we should have $3 \,^* \,3 = 3$.
However,using our defined operation,we get $3 \,^* \,3 = 3 + 3 = 6$.
Since $6 \neq 3$,the condition $a \,^* \,a = a$ does not hold for all binary operations on $N$.
Thus,the statement is false.