Show That the Sets \(A\times B\) and \(B\times A\) Have an Element in Common iff the Sets \(A\) and \(B\) Have an Element in Common
Question
Let \(A\) and \(B\) be two sets. Show that the sets \[ A\times B \] and \[ B\times A \] have an element in common if and only if the sets \[ A \] and \[ B \] have an element in common.
Solution
To Prove
\[ (A\times B)\cap(B\times A)\neq\phi \iff A\cap B\neq\phi \]
Proof
Suppose
\[ (A\times B)\cap(B\times A)\neq\phi \]
Then there exists an ordered pair \[ (x,y) \] such that
\[ (x,y)\in A\times B \]
and
\[ (x,y)\in B\times A \]
Since \[ (x,y)\in A\times B, \] we have
\[ x\in A \quad \text{and} \quad y\in B \]
Also, since \[ (x,y)\in B\times A, \] we have
\[ x\in B \quad \text{and} \quad y\in A \]
Therefore,
\[ x\in A\cap B \]
Hence,
\[ A\cap B\neq\phi \]
Conversely, suppose
\[ A\cap B\neq\phi \]
Then there exists an element \[ a \] such that
\[ a\in A \quad \text{and} \quad a\in B \]
Therefore,
\[ (a,a)\in A\times B \]
and also
\[ (a,a)\in B\times A \]
Hence,
\[ (a,a)\in (A\times B)\cap(B\times A) \]
Therefore,
\[ (A\times B)\cap(B\times A)\neq\phi \]
Hence proved that:
\[ \boxed{ (A\times B)\cap(B\times A)\neq\phi \iff A\cap B\neq\phi } \]