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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}$ in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.
✓

Answer

It is given that *: P(X) × P(X) → P(X) is defined as $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}\in\text{P(X)}$
We know that $\text{A}\cap\text{X}=\text{A}=\text{X}\cap\text{A}\forall\text{A}\in\text{P(X)}$
Thus, X is the identity element for the given binary operation *.
Now, an element $\text{A}\in\text{P(X)}$ is invertibleif there exists $\text{B}\in\text{P(X)}$ such that
A * B = X = B * A. (As X is the identity element)
i.e.,
$\text{A}\cap\text{B}=\text{X}=\text{B}\cap\text{A}$
This case is possible only when A = X = B.
Thus, X is the only invertible element in P(X) with respect to the given operation *.
Hence, the given result is proved.

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