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 6.3 - Page 372: 1

Answer

$counter\,\,example\,\,:\\ A=\left \{5,6 \right \},B=\left \{ 6,7 \right \},C=\left \{ 5,8 \right \}$

Work Step by Step

$For\,\,all\,\,sets\,\,A,\,B,\,and\,\,C\,\\ (A \cap B) \cup C = A \cap (B \cup C).\\ this\,\,not\,\,true:\\ counter\,\,example\,\,:\\ A=\left \{5,6 \right \},B=\left \{ 6,7 \right \},C=\left \{ 5,8 \right \}\\ A \cap B=\left \{5,6 \right \}\cap \left \{ 6,7 \right \}=\left \{ 6 \right \}\\ (A \cap B) \cup C=\left \{ 6 \right \}\cup \left \{ 5,8 \right \}=\left \{ 5,6,8 \right \} {\color{Red} (1)}\\ B\cup C=\left \{ 5,6,7,8 \right \}\\ A \cap (B \cup C)=\left \{ 5,6 \right \}\cap \left \{ 5,6,7,8 \right \}=\left \{ 5,6 \right \}{\color{Red} (2)} \\ from\,\,1,2\,\,(A \cap B) \cup C =\left \{ 5,6,8 \right \}\neq \left \{ 5,6 \right \}= A \cap (B \cup C)$
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