Question

Sets A, B, and Care subsets of the universal set U.These sets are defined as follows.U=(f.g. h. p. q, r, x, y, z)A={f, g, h, q,r}B={q, r, x, z}C={h, p, q,z}Find (A' n B') U C.Write your answer in roster form or as Ø.(A' n B') U C =

Answer 1

Given:

Sets A, B, and Care subsets of the universal set U.

These sets are defined as follows.

`\begin{gathered} U=\lbrace f,g,h,p,q,r,x,y,z\rbrace \\ A=\left\{f,g,h,q,r\right\} \\ B=\left\{q,r,x,z\right\} \\ C=\left\{h,p,q,z\right\} \end{gathered}`

Required:

Find

`(A^(\prime)\cap B^(\prime))UC.`

Step-by-step explanation:

The given sets are

`\begin{gathered} U=\lbrace f,g,h,p,q,r,x,y,z\rbrace \\ A=\left\{f,g,h,q,r\right\} \\ B=\left\{q,r,x,z\right\} \\ C=\left\{h,p,q,z\right\} \end{gathered}`
`\begin{gathered} A^(\prime)=U-A \\ =\lbrace f,g,h,p,q,r,x,y,z\rbrace-\left\{f,g,h,q,r\right\} \\ =\lbrace p,x,y,z\rbrace \end{gathered}`
`\begin{gathered} B^(\prime)=U-B \\ =\lbrace f,g,h,p,q,r,x,y,z\rbrace-\left\{q,r,x,z\right\} \\ =\lbrace f,g,h,p,y\rbrace \end{gathered}`
`\begin{gathered} A^(\prime)\cap B^(\prime)=\lbrace p,x,y,z\rbrace\cap\lbrace f,g,h,p,y\rbrace \\ =\lbrace p,y\rbrace \end{gathered}`
`\begin{gathered} (A^(\prime)\cap B^(\prime))\cup C=\lbrace p,y\rbrace\cup\left\{h,p,q,z\right\} \\ =\lbrace p,y,h,q,z\rbrace \end{gathered}`

Final Answer:

`\left(A^(\prime)\cap B^(\prime)\right)UC=\lbrace p,y,h,q,z\rbrace`