2020-06-22T08:38:46-04:00
Part (a): Let A and B are any sets then show that A-(A∩B)=(A∩A^c)∪(A∩B^c) by using membership table.
Part (b): Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions.
A∩B≠ϕ,B∩C≠ϕ,A∩C=ϕ,A⊈B,C⊈B.
1
2020-06-22T18:19:11-0400
(a):
A B A ∩ B A − ( A ∩ B ) 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 \def\arraystretch{1.5}
\begin{array}{c:c}
A & B & A\cap B & A-(A\cap B) \\ \hline
0 & 0 & 0 & 0 \\
0 &1 & 0 & 0 \\
1 & 0 & 0& 1 \\
\hdashline
1 &1 & 1 & 0 \\
\hdashline
\end{array} A 0 0 1 1 B 0 1 0 1 A ∩ B 0 0 0 1 A − ( A ∩ B ) 0 0 1 0
A B A C B C A ∩ A C A ∩ B C ( A ∩ A C ) ∪ ( A ∩ B C ) 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 \def\arraystretch{1.5}
\begin{array}{c:c}
A & B & A^C & B^C & A\cap A^C & A\cap B^C & (A\cap A^C)\cup (A\cap B^C) \\ \hline
0 & 0 & 1 & 1 & 0 & 0 & 0\\
\hdashline
0 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 1& 1 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
\end{array} A 0 0 1 1 B 0 1 0 1 A C 1 1 0 0 B C 1 0 1 0 A ∩ A C 0 0 0 0 A ∩ B C 0 0 1 0 ( A ∩ A C ) ∪ ( A ∩ B C ) 0 0 1 0
A − ( A ∩ B ) = ( A ∩ A C ) ∪ ( A ∩ B C ) A-(A\cap B)=(A\cap A^C)\cup (A\cap B^C) A − ( A ∩ B ) = ( A ∩ A C ) ∪ ( A ∩ B C )
(b)
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