Given a non-empty set X, consider the binary operation *: P(X) x P(X) → P (X) given by A * B = A ∩ B" A, B in 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 *

Solution:

A set is a well-defined collection of numbers, alphabets, objects, or any items. A subset is a part of the set

P(X) x P(X) → P (X) given by

A * B = A ∩ B" A, B in P (X)

A ∩ X = A

= X ∩ A for all A ∈ P (X)

⇒ A * X = A

= X * A for all A ∈ P (X)

X is the identity element for the given binary operation *.

An element A ∈ P (X) is invertible if there exists B ∈ P (X) such that

A * B = X

= B * A

[As X is the identity element]

Or

A ∩ B = X = B ∩ A

This case is possible only when A = X = B.

X is the only invertible element in P (X) with respect to the given operation *

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 9

Given a non-empty set X, consider the binary operation *: P(X) x P(X) → P (X) given by A * B = A ∩ B" A, B in 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 *.

Summary:

X is the only invertible element in P (X) with respect to the given operation *. An element A ∈ P (X) is invertible if there exists B ∈ P (X) such that A * B = X = B * A