If \(A=\{1,2,3\}\), \(B=\{4\}\), \(C=\{5\}\), Verify That \(A\times(B\cap C)=(A\times B)\cap(A\times C)\)
Question
If \[ A=\{1,2,3\},\quad B=\{4\},\quad C=\{5\}, \] verify that \[ A\times(B\cap C)=(A\times B)\cap(A\times C). \]
Solution
\[ B\cap C=\phi \]
\[ A\times(B\cap C)=A\times\phi=\phi \]
\[ A\times B= \{ (1,4),(2,4),(3,4) \} \]
\[ A\times C= \{ (1,5),(2,5),(3,5) \} \]
There is no common ordered pair.
\[ (A\times B)\cap(A\times C)=\phi \]
Thus,
\[ \boxed{ A\times(B\cap C)=(A\times B)\cap(A\times C) } \]