State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation $*$ on a set N, a $*$ a $=$ a $\forall a \in Q$
(ii) If $*$ is a commutative binary operation on N, then a$*$ (b$*$ c)$=$ (c $*$ b)$*$ a
(i) False.
A binary operation on N is defined as :
a $*$ a $=$ a $\forall a \in Q$. For example, a $*$ b$=$a + b $\forall a,b \in Q$ , then
a $*$ a $=$ a + a $=$ 2a$\ne$a.
Here $' * '$ is not defined.
(ii) True
a $*$(b $*$c)$=$(b $*$c)$*$a $=$(c $*$ b) $*$a. (b $*$c$=$c$*$ b is commutative)
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