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Question 9 - Miscellaneous - Chapter 1 Class 12 Relation and Functions

Last updated at May 29, 2023 by

Misc 13 - Let A * B = (A - B) U (B - A) - Chapter 1 Class 12

Misc 13 - Chapter 1 Class 12 Relation and Functions - Part 2
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Question 9 Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B − A), ∀ A, B ∈ P(X). Show that the empty set ϕ is the identity for the operation * and all the elements A of P(X) are invertible with A−1 = A. (Hint: (A − ϕ) ∪ (ϕ – A) = A and (A − A) ∪ (A − A) = A * A = ϕ). Identity e is the identity of * if a * e = e * a = a Here, A * ϕ = (A − ϕ) ∪ (ϕ – A) = A ∪ ϕ = A & ϕ * A = (ϕ − A) ∪ (A – ϕ) = ϕ ∪ A = A Since, A * ϕ = ϕ * A = A 𝛟 is the identity of operation * Invertible An element a in set is invertible if, there is an element in set such that , a * b = e = b * a Here, e = ϕ , b = A Now, A * A = (A − A) ∪ (A – A) = ϕ ∪ ϕ = ϕ & A * A = (A − A) ∪ (A – A) = ϕ ∪ ϕ = ϕ Since, A * A = ϕ = A * A Hence, all the elements A of P(X) are invertible with inverse of A = A

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.