Let \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{5,6\}\), Verify That \(A\times(B\cap C)=(A\times B)\cap(A\times C)\)
Question
Let \[ A=\{1,2\},\quad B=\{1,2,3,4\},\quad C=\{5,6\} \] verify that \[ A\times(B\cap C)=(A\times B)\cap(A\times C). \]
Solution
\[ B\cap C=\phi \]
\[ A\times(B\cap C)=A\times\phi=\phi \]
\[ A\times B= \{ (1,1),(1,2),(1,3),(1,4), \]
\[ (2,1),(2,2),(2,3),(2,4) \} \]
\[ A\times C= \{ (1,5),(1,6),(2,5),(2,6) \} \]
There is no common ordered pair.
\[ (A\times B)\cap(A\times C)=\phi \]
Thus,
\[ \boxed{ A\times(B\cap C)=(A\times B)\cap(A\times C) } \]