Check sibling questions

Misc 10 - Show that A B = A C need not imply B = C - Chapter 1

Misc 10 - Chapter 1 Class 11 Sets - Part 2

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Misc 10 Show that A ∩ B = A ∩ C need not imply B = C. We have to prove false, so we take a example It is given that A ∩ B = A ∩ C i.e. Common element in set A & B = Common element in set A & C Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5} A ∩ B = {0} and A ∩ C = {0} Here, A ∩ B = A ∩ C = {0} But B ≠ C as 2 is in set B, but not in A Hence proved Misc 11 Let A and B be sets. If A ∩ X = B ∩ X = ∅ and A ∪ X = B ∪ X for some set X, show that A = B. (Hints: A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law) Given: Let A and B be two sets such that A ∩ X = B ∩ X = ∅ and A ∪ X = B ∪ X for some set X. To prove: A = B Proof: Let A = A ∩ (A ∪ X) A = A ∩ (B ∪ X) (Given A ∪ X = B ∪ X) Let A = A ∩ (A ∪ X) Given A ∪ X = B ∪ X A = A ∩ (B ∪ X) Using distributive law : A ∩ (B ∪ C)= (A ∩ B) ∪ (A ∩ C) = (A ∩ B) ∪ (A ∩ X) As A ∩ X = ∅ given = (A ∩ B) ∪ ∅ A = A ∩ B Let B = B ∩ (B ∪ X) Given A ∪ X = B ∪ X B = B ∩ (A ∪ X) Using distributive law: A ∪ (B ∩ C)= (A ∩ B) ∪ (A ∩ C) = (B ∩ A) ∪ (B ∩ X) As B ∩ X = Φ = (B ∩ A) ∪ Φ B = B ∩ A B = A ∩ B From (1) and (2), A = A ∩ B & B = A ∩ B ∴ A = B Hence proved

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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.