Respuesta :
Given: Sets B and C are subsets of the universal set U.
These sets are defined as follows-
[tex]\begin{gathered} U=\left\{1,3,5,6,7\right\} \\ B=\left\{1,3,6\right\} \\ C=\left\{1,3,5\right\} \end{gathered}[/tex]Required: To determine the following sets-
[tex]\begin{gathered} B^{\prime}\cup C^{\prime} \\ B^{\prime}\cap C \end{gathered}[/tex]Explanation: The complement of a set A with the universal set U is defined as-
[tex]A^{\prime}=U-A[/tex]Hence, the complement of set B is-
[tex]\begin{gathered} B^{\prime}=U-B \\ =\left\{1,3,5,6,7\right\}-\left\{1,3,6\right\} \\ =\lbrace5,7\rbrace \end{gathered}[/tex]Similarly, the complement of set C is-
[tex]\begin{gathered} C^{\prime}=\left\{1,3,5,6,7\right\}-\left\{1,3,5\right\} \\ =\lbrace6,7\rbrace \end{gathered}[/tex]Now,
[tex]\begin{gathered} B^{\prime}\cup C^{\prime}=\lbrace5,7\rbrace\cup\lbrace6,7\rbrace \\ =\lbrace5,6,7\rbrace \end{gathered}[/tex]Similarly-
[tex]\begin{gathered} B^{\prime}\cap C=\lbrace5,7\rbrace\cap\left\{1,3,5\right\} \\ =\lbrace5\rbrace \end{gathered}[/tex]Final Answer: (a)-
[tex]B^{\prime}\cup C^{\prime}=\lbrace5,6,7\rbrace[/tex](b)-
[tex]B^{\prime}\cap C=\lbrace5\rbrace[/tex]