State Whether the Statement is True or False: If \(P=\{m,n\}\) and \(Q=\{n,m\}\), Then \(P\times Q=\{(m,n),(n,m)\}\)
Question
State whether the following statement is true or false. If the statement is false, rewrite the statement correctly:
If \[ P=\{m,n\} \] and \[ Q=\{n,m\}, \] then \[ P\times Q=\{(m,n),(n,m)\}. \]
Solution
Given:
\[ P=\{m,n\} \]
and
\[ Q=\{n,m\}=\{m,n\} \]
Now form the Cartesian product \(P\times Q\).
All possible ordered pairs are:
\[ (m,m),\ (m,n),\ (n,m),\ (n,n) \]
Therefore,
\[ P\times Q= \{(m,m),(m,n),(n,m),(n,n)\} \]
Hence, the given statement is:
\[ \boxed{\text{False}} \]
Correct Statement
\[ \boxed{ P\times Q= \{(m,m),(m,n),(n,m),(n,n)\} } \]