Q. 145.0( 1 Vote )

| Let * be a bina

Answer :

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.


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