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 6.1 - Page 351: 35

Answer

$a.\,\,=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \}\\ b.\,\,=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \} \\ c.\,\,=\left \{ (a,2),(b,2) \right \}\\ d.\,\,=\left \{ (a,2),(b,2) \right \}\\$

Work Step by Step

$A=\left \{ a,b \right \},B=\left \{ 1,2 \right \},C=\left \{ 2,3 \right \} \\ {\color{Red} a.\,\,}B\cup C=\left \{ 1,2 \right \}\cup \left \{ 2,3 \right \}=\left \{ 1,2,3 \right \} \\ so\,\,A\times (B\cup C)=\left \{ a,b \right \}\times \left \{ 1,2,3 \right \} =\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \} \\ {\color{Red} b.\,\,}A\times B=\left \{ a,b \right \}\times \left \{ 1,2 \right \}=\left \{ (a,1),(a,2),(b,1),(b,2) \right \} \\ A\times C=\left \{ a,b \right \}\times \left \{ 2,3 \right \}=\left \{ (a,2),(a,3),(b,2),(b,3) \right \}\\ (A\times B)\cup(A\times C)=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \}\\ {\color{Red} c.\,\,} B\cap C=\left \{ 1,2 \right \}\cap \left \{ 2,3 \right \}=\left \{ 2 \right \}\\ so\,\,A\times (B\cap C)=\left \{ a,b \right \}\times \left \{ 2 \right \}=\left \{ (a,2),(b,2) \right \}\\ {\color{Red} d.\,\,} (A\times B)\cap (A\times C)=\left \{ (a,1),(a,2),(b,1),(b,2) \right \}\times \left \{(a,2),(a,3),(b,2),(b,3) \right \}\\$ $(A\times B)\cap (A\times C)=\left \{ (a,2),(b,2) \right \} $
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