Let \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{5,6\}\) and \(D=\{5,6,7,8\}\), Verify That \(A\times C \subset B\times D\)
Question
Let \[ A=\{1,2\},\quad B=\{1,2,3,4\}, \]
\[ C=\{5,6\},\quad D=\{5,6,7,8\} \]
Verify that \[ A\times C \subset B\times D. \]
Solution
\[ A\times C= \{ (1,5),(1,6), \]
\[ (2,5),(2,6) \} \]
\[ B\times D= \{ (1,5),(1,6),(1,7),(1,8), \]
\[ (2,5),(2,6),(2,7),(2,8), \]
\[ (3,5),(3,6),(3,7),(3,8), \]
\[ (4,5),(4,6),(4,7),(4,8) \} \]
Every ordered pair of \[ A\times C \] belongs to \[ B\times D. \]
Therefore,
\[ \boxed{ A\times C \subset B\times D } \]