Question

Sets B and C are subsets of the universal set U.These sets are defined as follows.U= { 1, 3, 5, 6, 7}B={ 1, 3, 6}C= { 1, 3, 5 }Find the following sets.Write your answer in roster form or as Ø.(a) B' Ụ C' = (b) B' n C =

Answer 1

Given: Sets B and C are subsets of the universal set U.

These sets are defined as follows-

`\begin{gathered} U=\left\{1,3,5,6,7\right\} \\ B=\left\{1,3,6\right\} \\ C=\left\{1,3,5\right\} \end{gathered}`

Required: To determine the following sets-

`\begin{gathered} B^(\prime)\cup C^(\prime) \\ B^(\prime)\cap C \end{gathered}`

Explanation: The complement of a set A with the universal set U is defined as-

`A^(\prime)=U-A`

Hence, the complement of set B is-

`\begin{gathered} B^(\prime)=U-B \\ =\left\{1,3,5,6,7\right\}-\left\{1,3,6\right\} \\ =\lbrace5,7\rbrace \end{gathered}`

Similarly, the complement of set C is-

`\begin{gathered} C^(\prime)=\left\{1,3,5,6,7\right\}-\left\{1,3,5\right\} \\ =\lbrace6,7\rbrace \end{gathered}`

Now,

`\begin{gathered} B^(\prime)\cup C^(\prime)=\lbrace5,7\rbrace\cup\lbrace6,7\rbrace \\ =\lbrace5,6,7\rbrace \end{gathered}`

Similarly-

`\begin{gathered} B^(\prime)\cap C=\lbrace5,7\rbrace\cap\left\{1,3,5\right\} \\ =\lbrace5\rbrace \end{gathered}`

Final Answer: (a)-

`B^(\prime)\cup C^(\prime)=\lbrace5,6,7\rbrace`

(b)-

`B^(\prime)\cap C=\lbrace5\rbrace`