If \(A=\{1,2,3\}\), \(B=\{4\}\), \(C=\{5\}\), Verify That \(A\times(B\cup C)=(A\times B)\cup(A\times C)\)
Question
If \[ A=\{1,2,3\},\quad B=\{4\},\quad C=\{5\}, \] verify that \[ A\times(B\cup C)=(A\times B)\cup(A\times C). \]
Solution
\[ B\cup C=\{4,5\} \]
\[ A\times(B\cup C)= \{ (1,4),(1,5), \]
\[ (2,4),(2,5), \]
\[ (3,4),(3,5) \} \]
\[ A\times B= \{ (1,4),(2,4),(3,4) \} \]
\[ A\times C= \{ (1,5),(2,5),(3,5) \} \]
\[ (A\times B)\cup(A\times C)= \{ (1,4),(1,5), \]
\[ (2,4),(2,5), \]
\[ (3,4),(3,5) \} \]
Thus,
\[ \boxed{ A\times(B\cup C)=(A\times B)\cup(A\times C) } \]