Let * be any binary operation on the set R defined by a * b = a + b – ab, then the binary operation * isa)Associativeb)Commutative but not associativec)Commutative and associatived)CommutativeCorrect answer is option 'C'. Can you explain this answer?

Question

Let * be any binary operation on the set R defined by a * b = a + b &#8211; ab, then the binary operation * is&#8203;a)Associativeb)Commutative but not associativec)Commutative and associatived)CommutativeCorrect answer is option 'C'. Can you explain this answer?

Accepted Answer

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}’.

Answer

a * b = 1 + ab, a, b ∈ R.

Answer

Understanding the Operation
The binary operation * on the set R is defined as:
- a * b = a + b + ab
To analyze this operation, we will check for commutativity and associativity.
Commutativity
An operation is commutative if a * b = b * a for all a, b in R.
- Let’s evaluate a * b:
- a * b = a + b + ab
- Now, evaluate b * a:
- b * a = b + a + ba
Since addition and multiplication are commutative:
- a + b = b + a
- ab = ba
Thus, a * b = b * a, proving that the operation * is commutative.
Associativity
An operation is associative if (a * b) * c = a * (b * c) for all a, b, c in R.
- Calculate (a * b) * c:
- a * b = a + b + ab
- Let x = a * b, then x * c = (a + b + ab) * c
- = (a + b + ab) + c + (a + b + ab)c
- Calculate a * (b * c):
- b * c = b + c + bc
- Then a * (b * c) = a + (b + c + bc) + a(b + c + bc)
After simplification, we find that both expressions yield the same result, confirming that the operation * is associative.
Conclusion
The binary operation * is therefore:
- Commutative
- Associative
This leads us to the correct answer: option C: Commutative and associative.

Top Courses for JEEView all

Physics for JEE Main & Advanced

Chapter-wise Tests for JEE Main & Advanced

Mock Tests for JEE Main and Advanced 2026

Chemistry for JEE Main & Advanced

Mathematics (Maths) for JEE Main & Advanced

Top Courses for JEE

Top Courses for JEE

Related Content

JEE Courses

Mock Test Series

Past Year Papers

Additional Courses

Company