Discrete Mathematics with Applications 4th Edition

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

Chapter 7 - Functions - Exercise Set 7.1 - Page 396: 42

Answer

$F(A \cap B) \subseteq F(A) \cap F(B) \\ let\,\,y\in F(A \cap B)\Rightarrow y=F(x)\,for\,some\,x\in A\cap B\\ x\in A\cap B\Rightarrow x\in A\,and\,x\in B \\ so\,\,y=F(x)\\ for\,some\,x\in A \Rightarrow y\in F(A)\\ for\,some\,x\in B \Rightarrow y\in F(B)\\ \because y\in F(A)\,and\,y\in F(B)\\ \therefore y\in F(A)\cap F(B)\\ \because y\in F(A\cap B)\Rightarrow y\in F(A)\cap F(B)\\ \therefore F(A \cap B) \subseteq F(A) \cap F(B) \\ $

Work Step by Step

$F(A \cap B) \subseteq F(A) \cap F(B) \\ let\,\,y\in F(A \cap B)\Rightarrow y=F(x)\,for\,some\,x\in A\cap B\\ x\in A\cap B\Rightarrow x\in A\,and\,x\in B \\ so\,\,y=F(x)\\ for\,some\,x\in A \Rightarrow y\in F(A)\\ for\,some\,x\in B \Rightarrow y\in F(B)\\ \because y\in F(A)\,and\,y\in F(B)\\ \therefore y\in F(A)\cap F(B)\\ \because y\in F(A\cap B)\Rightarrow y\in F(A)\cap F(B)\\ \therefore F(A \cap B) \subseteq F(A) \cap F(B) \\ $
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