Question

State whether the following statements are true of false. Justify.
(i) For an arbitrary binary operation $\ast$ on as set $N,a\ast a=a\forall aϵN$
(ii) If $\ast$ is a commutative binary operation on $N$, then $a\ast (b\ast c)=(c\ast b)\ast a$

• A. True
• B. False

Answer 1

The correct option is A True

i)Given $a\ast a=a\forall aϵN$ on a set $N$ for an arbitary binary operation $\ast$.

The binary operation can be defined as,$a\times b=a^2$

Clearly $a^2\ne a\forall aϵA$.Therefore the statement is false.

ii)Given binary operator $\ast$ on set $N$.We need to prove that $a\ast (b\ast a)=(c\ast b)\ast a$

An operation $\ast$ on $A$ is commutative

if $a\ast b=b\ast a\forall a,bϵA$

$\therefore a\ast (b\ast a)=(c\ast b)\ast a$

since $\ast$ is commutative.

$\therefore$the statement is true.