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# State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation $\ast$ on a set N, $a \ast a=a \forall a \in N$

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• A binary operation $\ast$ on a set $A$ is a function $\ast$ from $A \times A$ to $A$. Therefore, if $a,b \in A \Rightarrow a \ast b \in A\; \forall\; a,b, \in A$
Given $a \ast a=a \forall a \in N$ on a set $N$ for an arbitrary binary operation $\ast$:
Let us define the binary operation as $a \ast b = a^2$. Clearly, $a^2 \neq a \forall a \in A$. Therefore the statement is false.
We can verify this easily. Let $a=b=2 \rightarrow a \ast b = 2^2 = 4$ which is $\neq a = 2$.
If  *  is a binary operation  and commutative then  a*(b*c)=(c*b)*a  is true
Because, a*(b*c)=a*(c*b)   (  since * is commutative)
Again using the same commutative property,
a*(c*b)=(c*b)*a