If \(A=\{2,3\}\), \(B=\{4,5\}\), \(C=\{5,6\}\), Find \(A\times(B\cup C)\), \(A\times(B\cap C)\) and \((A\times B)\cup(A\times C)\)
Question
If \[ A=\{2,3\},\quad B=\{4,5\},\quad C=\{5,6\}, \] find:
\[ A\times(B\cup C), \]
\[ A\times(B\cap C), \]
and
\[ (A\times B)\cup(A\times C). \]
Solution
\[ B\cup C=\{4,5,6\} \]
\[ A\times(B\cup C)= \{ (2,4),(2,5),(2,6), \]
\[ (3,4),(3,5),(3,6) \} \]
\[ B\cap C=\{5\} \]
\[ A\times(B\cap C)= \{ (2,5),(3,5) \} \]
\[ A\times B= \{ (2,4),(2,5),(3,4),(3,5) \} \]
\[ A\times C= \{ (2,5),(2,6),(3,5),(3,6) \} \]
Therefore,
\[ (A\times B)\cup(A\times C)= \{ (2,4),(2,5),(2,6), \]
\[ (3,4),(3,5),(3,6) \} \]