Question

See attached question Are the statements equal for all sets A, b, and c ?Option yes or no

Answer 1

The first statement is:

`(A\cup B)^(\prime)\cap C`

Then, first we need to find (AUB)', so:

`\begin{gathered} (A\cup B)=\lbrace I,II,III,IV,V,VI\rbrace \\ (A\cup B)^(\prime)=\lbrace VII,VIII\rbrace \end{gathered}`

Now that we know (AUB)', let's find (AUB)'nC:

`(A\cup B)^(\prime)\cap C=\lbrace VII\rbrace`

The second statement is:

`(A^(\prime)\cup C^(\prime))\cap(B^(\prime)\cup C)`

First, let's find (A'UC'):

`\begin{gathered} A=\lbrace I,II,IV,V\rbrace \\ A^(\prime)=\lbrace III,VI,VII,VIII\rbrace \\ C=\lbrace IV,V,VI,VII\rbrace \\ C^(\prime)=\lbrace I,II,III,VIII\rbrace \\ (A^(\prime)\cup C^(\prime))=\lbrace I,II,III,VI,VII,VIII\rbrace \end{gathered}`

Now, let's find (B'UC):

`\begin{gathered} B=\lbrace II,III,V,VI\rbrace \\ B^(\prime)=\lbrace I,IV,VII,VIII\rbrace \\ (B^(\prime)\cup C)=\lbrace I,IV,V,VI,VII,VIII\rbrace \end{gathered}`

Finally, let's find (A'UC')n(B'UC):

`(A^(\prime)\cup C^(\prime))\cap(B^(\prime)\cup C)=\lbrace I,VI,VII,VIII\rbrace`

Then, the answer is the statements are NOT equal for all sets A, B and C.