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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
Misc 13 - Chapter 1 Class 12 Relation and Functions - Part 3

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Misc 13 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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.