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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Find which of the binary operations are commutative and which are associative.
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)
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Answer

A = N × N and * is a binary operation defined on A.
(a, b) * (c, d) = (a + b, c + d) = (c + a, d + b) = (c, d) * (a, b)
$\therefore$ The operation is commutative
Again, [(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f) = (a + c + e, b + d + f)
And (a, b)[(c, d) * (e, f)] = (a, b) * (c + e, e + f) = (a + c + e, b + d + f)
Here, [(a, b) * (c, d)] * (e, f) = (a, b)[(c, d) * (e, f)]
$\therefore$ The operation is associative.
Let the identity function be (e, f), then (a, b) * (e, f) = (a + e, b + f)
For identity function a = a + e $\Rightarrow\ \ \text{e}=0$
And for b + f = b $\Rightarrow\ \ \text{f}=0$
As $0\neq\text{N},$ therefore identity element does not exist.

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