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Misc 9 - Consider binary operation *, given by A *B = AB, P(X) is powe

Misc 9 - Chapter 1 Class 12 Relation and Functions - Part 2
Misc 9 - Chapter 1 Class 12 Relation and Functions - Part 3

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Misc 9 (Introduction) Given a non-empty set X, consider the binary operation *: P(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*. Taking an example Let X = {1, 2, 3} P(X) = Power set of X = Set of all subsets of X = { 𝜙 , {1} , {2} , {3}, {1, 2} , {2, 3} , {1, 3}, {1, 2, 3} } A * X = A ∩ X = A X * A = X ∩ A = A Misc 9 Given a non-empty set X, consider the binary operation *: P(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*. Identity e is the identity of * if a * e = e * a = a A * X = A ∩ X = A X * A = X ∩ A = A So, A * X = A = X * A , for all A ∈ P (X) Thus, X is the identity element for the given binary 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 = X So, A * B = X = B * A i.e. A ∩ B = X This is only possible if A = B = X So, A * X = A = X * A , for all A ∈ P (X) Thus, X is the only invertible element in P(X) with respect to the given operation*.

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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.