| Let * be a binary operation on Q defined by

Show that * is commutative as well as associative. Also find its identity elements, if it exists.

We know that, a binary operation is commutative if changing the order of the operands does not change the result. It should satisfy this condition,

a*b = b*a …(i)

We are given that,

Take Left Hand Side (LHS) of (i),

LHS = a*b

Take Right Hand Side (RHS) of (i),

The binary operation defined would remain same but since RHS has b*a instead of a*b, then a and b in the operation would change its position correspondingly.

RHS = b*a

Even after changing positions of a and b, the equation remains same.

So, LHS = RHS.

⇒ a*b = b*a

Hence, binary operation * is commutative.

We know that, the associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. It should satisfy this condition,

(a*b)*c = a*(b*c) …(ii)

We are given that,

…(iii)

Take LHS of (ii),

LHS = (a*b)*c

We will solve this by BODMAS rule. That is, brackets will be solved first.

Using result of (iii), we get

Now, replace a by and b by c in (iii), we get

Take RHS of (ii),

RHS = a*(b*c)

Let us solve this by BODMAS rule. That is, brackets will be solved first.

Replacing a by b and b by c in (iii), we get

Now, replacing a by a and b by in (iii), we get

Since, LHS = RHS.

Hence, binary operation * is associative.

We need to find an identity element.

We know that, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

So, let e ∈ Q be the identity element.

Then,

a*e = e*a = a

Let us find e.

Take a*e = a

Replace a by a and b by e in (iii), we get

Or we could take e*a = a, the value of e would be same.

Therefore, its identity element is 5/3.