The first statement is:
[tex](A\cup B)^{\prime}\cap C[/tex]Then, first we need to find (AUB)', so:
[tex]\begin{gathered} (A\cup B)=\lbrace I,II,III,IV,V,VI\rbrace \\ (A\cup B)^{\prime}=\lbrace VII,VIII\rbrace \end{gathered}[/tex]Now that we know (AUB)', let's find (AUB)'nC:
[tex](A\cup B)^{\prime}\cap C=\lbrace VII\rbrace[/tex]The second statement is:
[tex](A^{\prime}\cup C^{\prime})\cap(B^{\prime}\cup C)[/tex]First, let's find (A'UC'):
[tex]\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}[/tex]Now, let's find (B'UC):
[tex]\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}[/tex]Finally, let's find (A'UC')n(B'UC):
[tex](A^{\prime}\cup C^{\prime})\cap(B^{\prime}\cup C)=\lbrace I,VI,VII,VIII\rbrace[/tex]Then, the answer is the statements are NOT equal for all sets A, B and C.
