Test: Binary Operations - JEE MCQ

Correct Answer :- a

Explanation : No. of elements : 3

Total no.of binary operations on a set consisting 'n' elements is given by n^{n^2}

= So the total no.of binary sets is 3⁹.

 a ∗ b = e = b ∗a,
(0,0) (1,1,) (2,2) (3,3) (4,4) (5,5) (6,6)

Commutative law, in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + b = b + a and ab = ba

 n(A) = n^{n^2}
=2^{2^2}
= 2⁴
= 16

We have ∗ is defined on the set {1,2,3,4,5}  by a∗b = LCM of a and b
Now, 3∗4 = LCM of 3 and 4=12
But, 12 does not belong to {1,2,3,4,5}.
Hence, ′∗′ is not a binary operation.

Given that ο is a binary operation on Q – { – 1} defined by aοb = a + b – ab for all a,b∈Q – { – 1}.
We know that commutative property is pοq = qοp, where ο is a binary operation.
Let’s check the commutativity of given binary operation:
⇒ aοb = a + b – ab
⇒ bοa = b + a – ba = a + b – ab
⇒ b*a = a*b
∴ Commutative property holds for given binary operation ‘ο’ on ‘Q – [ – 1]’.
We know that associative property is (pοq)οr = pο(qοr)
Let’s check the associativity of given binary operation:
⇒ (aοb)οc = (a + b – ab)οc
⇒ (aοb)οc = a + b – ab + c – ((a + b – ab)×c)
⇒ (aοb)οc = a + b + c – ab – ac – ab + abc ...... (1)
⇒ aο(bοc) = aο(b + c – bc)
⇒ aο(bοc) = a + b + c – bc – (a×(b + c – bc))
⇒ a*(b*c) = a + b + c – ab – bc – ac + abc ...... (2)
From (1) and (2)
⇒ (aob)oc = ao(boc)
Hence,we can clearly say that associativity hold for the given binary operation on ‘Q – { – 1}’.
Given that ο is a binary operation on Q – { – 1} defined by aοb = a + b – ab for all a,b∈Q – { – 1}.
We know that commutative property is pοq = qοp, where ο is a binary operation.
Let’s check the commutativity of given binary operation:
⇒ aοb = a + b – ab
⇒ bοa = b + a – ba = a + b – ab
⇒ b*a = a*b
∴ Commutative property holds for given binary operation ‘ο’ on ‘Q – [ – 1]’.
We know that associative property is (pοq)οr = pο(qοr)
Let’s check the associativity of given binary operation:
⇒ (aοb)οc = (a + b – ab)οc
⇒ (aοb)οc = a + b – ab + c – ((a + b – ab)×c)
⇒ (aοb)οc = a + b + c – ab – ac – ab + abc ...... (1)
⇒ aο(bοc) = aο(b + c – bc)
⇒ aο(bοc) = a + b + c – bc – (a×(b + c – bc))
⇒ a*(b*c) = a + b + c – ab – bc – ac + abc ...... (2)
From (1) and (2)
⇒ (aob)oc = ao(boc)
Hence,we can clearly say that associativity hold for the given binary operation on ‘Q – { – 1}’.

a*e = a = e*a
ae/2 = a
e = 2