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Question
Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
Options
Commutative but not associative
Associative but not commutative
Neither commutative nor associative
Both commutative and associative
Solution
Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is commutative but not associative.
Explanation:
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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