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Question
Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:
| * | a | b | c | d |
| a | a | c | b | d |
| b | d | a | b | c |
| c | c | d | a | a |
| d | d | b | a | c |
Is it commutative and associative?
Solution
From the table
b * c = b
c * b = d
So, the binary operation is not commutative.
To check whether the given operation is associative.
Let a, b, c ∈ A.
To prove the associative property we have to prove that a * (b * c) = (a * b) * c
From the table,
L.H.S: b * c = b
So, a * (b * c) = a * b = c ........(1)
R.H.S: a * b = c
So, (a * b) * c = c * c = a ........(2)
(1) ≠ (2).
So, a * (b * c) ≠ (a * b) * c
∴ The binary operation is not associative.
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