Question

Let * be a binary operation on the set $Q$ of rational numbers as follows:
$a\ast b=a-b$
Find which of the binary operations are commutative and which are associative.

Answer 1

Consider given the given binary operation,

$a\ast b=a-b$

Commutative –

$a\ast b=a-b$

$b\ast a=b-a$

$a-b\ne b-a$

$a\ast b\ne b\ast a$

Hence, given operation is not commutative.

Associative-

$a\ast b=a-b$

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

$=a-(b-c)=a-b+c$

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

$=a-b-c$

Now,

$a\ast (b\ast c)\ne (a\ast b)\ast c$

$\because a-b+ca\ne a-b-c$

Hence, given operation is not associative.