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.

(a):

$\def\arraystretch{1.5} \begin{array}{c:c} A & B & A\cap B & A-(A\cap B) \\ \hline 0 & 0 & 0 & 0 \\ 0 &1 & 0 & 0 \\ 1 & 0 & 0& 1 \\ \hdashline 1 &1 & 1 & 0 \\ \hdashline \end{array}$

$\def\arraystretch{1.5} \begin{array}{c:c} A & B & A^C & B^C & A\cap A^C & A\cap B^C & (A\cap A^C)\cup (A\cap B^C) \\ \hline 0 & 0 & 1 & 1 & 0 & 0 & 0\\ \hdashline 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1& 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$

$A-(A\cap B)=(A\cap A^C)\cup (A\cap B^C)$

(b)

(a):

$\def\arraystretch{1.5} \begin{array}{c:c} A & B & A\cap B & A-(A\cap B) \\ \hline 0 & 0 & 0 & 0 \\ 0 &1 & 0 & 0 \\ 1 & 0 & 0& 1 \\ \hdashline 1 &1 & 1 & 0 \\ \hdashline \end{array}$

$\def\arraystretch{1.5} \begin{array}{c:c} A & B & A^C & B^C & A\cap A^C & A\cap B^C & (A\cap A^C)\cup (A\cap B^C) \\ \hline 0 & 0 & 1 & 1 & 0 & 0 & 0\\ \hdashline 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1& 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$

$A-(A\cap B)=(A\cap A^C)\cup (A\cap B^C)$

(b)