Answer
$a.\,\,=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \}\\
b.\,\,=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \} \\
c.\,\,=\left \{ (a,2),(b,2) \right \}\\
d.\,\,=\left \{ (a,2),(b,2) \right \}\\$
Work Step by Step
$A=\left \{ a,b \right \},B=\left \{ 1,2 \right \},C=\left \{ 2,3 \right \} \\
{\color{Red} a.\,\,}B\cup C=\left \{ 1,2 \right \}\cup \left \{ 2,3 \right \}=\left \{ 1,2,3 \right \} \\
so\,\,A\times (B\cup C)=\left \{ a,b \right \}\times \left \{ 1,2,3 \right \} =\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \} \\
{\color{Red} b.\,\,}A\times B=\left \{ a,b \right \}\times \left \{ 1,2 \right \}=\left \{ (a,1),(a,2),(b,1),(b,2) \right \} \\
A\times C=\left \{ a,b \right \}\times \left \{ 2,3 \right \}=\left \{ (a,2),(a,3),(b,2),(b,3) \right \}\\
(A\times B)\cup(A\times C)=\left \{ (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) \right \}\\
{\color{Red} c.\,\,} B\cap C=\left \{ 1,2 \right \}\cap \left \{ 2,3 \right \}=\left \{ 2 \right \}\\
so\,\,A\times (B\cap C)=\left \{ a,b \right \}\times \left \{ 2 \right \}=\left \{ (a,2),(b,2) \right \}\\
{\color{Red} d.\,\,} (A\times B)\cap (A\times C)=\left \{ (a,1),(a,2),(b,1),(b,2) \right \}\times \left \{(a,2),(a,3),(b,2),(b,3) \right \}\\$
$(A\times B)\cap (A\times C)=\left \{ (a,2),(b,2) \right \}
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