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Let * be binary operation defined

Question:

Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is

(i) commutative but not associative

(ii) associative but not commutative

(iii) neither commutative nor associative

(iv) both commutative and associative

Solution:

(i) Given that * is a binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R

So, we have a * b = ab + 1 = b * a

So, * is a commutative binary operation.

Now, a * (b * c) = a * (1 + bc) = 1 + a (1 + bc) = 1 + a + abc

Also,

(a * b) * c = (1 + ab) * c = 1 + (1 + ab) c = 1 + c + abc

Thus, a * (b * c) ≠ (a * b) * c

Hence, * is not associative.

Therefore, * is commutative but not associative.

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