Given \(A=\{1,2,3\}\), \(B=\{3,4\}\), \(C=\{4,5,6\}\), Find \((A\times B)\cap(B\times C)\)
Question
Given \[ A=\{1,2,3\}, \] \[ B=\{3,4\}, \] and \[ C=\{4,5,6\}, \] find \[ (A\times B)\cap(B\times C). \]
Solution
\[ A\times B= \{ (1,3),(1,4), (2,3),(2,4), (3,3),(3,4) \} \]
\[ B\times C= \{ (3,4),(3,5),(3,6), (4,4),(4,5),(4,6) \} \]
The common ordered pair is:
\[ (3,4) \]
Therefore,
\[ \boxed{ (A\times B)\cap(B\times C)=\{(3,4)\} } \]