Question

For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.

(ii) On, $Q^{+}$ define $a\ast b=ab+1$

Answer 1

Checking for binary.
Given: On $Q$, $a\ast b=ab+1$
$\forall a,b\in Q,ab+1\in Q$
So, ∗ is binary.
Checking for commutative.
Given: On $Q$, $a\ast b=ab+1$
∗ is commutative if,
$a\ast b=b\ast a$
Now,
$a\ast b=ba+1$
And,
$b\ast a=ba+1$
$ab+1$
Since, $a\ast b=b\ast a$ $\forall a,b\in Q$
∗ is commutative.
Checking for associative.
∗ is associative if,
$(a\ast b)\ast c=a^{\ast }\left(b^{\ast }c\right)$
Now,
$=(ab+1)c+1$
$=(abc+c+1$)
And,
$(a\ast b)\ast c=a^{\ast }(bc+1)$
$=a(bc+1)+1$
Since, $=abc+a+1$ $(a\ast b)\ast c\ne a^{\ast }(b\ast c)$
∗ is not an associative binary operation