Misc 11 - Chapter 1 Class 11 Sets

Transcript

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) 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) B = B (A X) Using distributive law :A (B C)= (A B) (A C) = (B A) (B X) As B X = = (B A) B = B A B = A B From (1) and (2), A = A B & B = A B A = B Hence proved