Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 6 - Set Theory - Exercise Set 6.2 - Page 365: 13

Answer

$ For\,\,all\,\,sets\,A, B, and\,\,C\\ if\,A\subseteq B\,\,then\,A\cap C\subseteq B\cap C \\ proof:\\ x\in A\cap C\ (by\,\,def.of\,\,inter\! section\,of\,sets)\\ x\in A\,\,and\,\,x\in c\,, \,\,\,\because A\subseteq B \\ x\in B\,\,and\,\,x\in c(by\,\,def.of\,\,inter\! section\,of\,sets) \\ \therefore x\in B\cap C\\ \therefore\,\,A\cap C\subseteq B\cap C $

Work Step by Step

$ For\,\,all\,\,sets\,A, B, and\,\,C\\ if\,A\subseteq B\,\,then\,A\cap C\subseteq B\cap C \\ proof:\\ x\in A\cap C\ (by\,\,def.of\,\,inter\! section\,of\,sets)\\ x\in A\,\,and\,\,x\in c\,, \,\,\,\because A\subseteq B \\ x\in B\,\,and\,\,x\in c(by\,\,def.of\,\,inter\! section\,of\,sets) \\ \therefore x\in B\cap C\\ \therefore\,\,A\cap C\subseteq B\cap C $
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