Question

State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation * on a set N , a * a = a a * N . (ii) If * is a commutative binary operation on N , then a * ( b * c ) = ( c * b ) * a

Answer 1

(i)

It is given that a∗a=a;∀a∈N.

Assume an operation ∗ is defined on N as a*b=a+b where a,b∈N.

Consider the random value, b=a=4.

a∗b=a∗a =4∗4 =4+4 =8

As the value of the statement a∗a is not equal to a, so the statement is false.

Thus, the statement ( i ) is false.

(ii)

An operation ∗ is defined on N as, a∗( b∗c )=( c∗b )∗a, where a,b∈N.

Right hand side of the above equation is written as,

( c∗b )∗a=a∗( c∗b ) =a∗( b∗c )

It shows that left side is equal to the right side, so it is commutative.

As the above statement proves that ∗ is a binary operation on N.

Thus, the statement ( ii ) is true.