State Whether the Statement is True or False: If \(A=\{1,2\}\), \(B=\{3,4\}\), Then \(A\times(B\cap\Phi)=\Phi\)
Question
State whether the following statement is true or false. If the statement is false, rewrite the statement correctly:
If \[ A=\{1,2\},\quad B=\{3,4\}, \] then \[ A\times(B\cap\Phi)=\Phi \]
Solution
First find:
\[ B\cap\Phi \]
The intersection of any set with the empty set is the empty set.
Therefore,
\[ B\cap\Phi=\Phi \]
Now,
\[ A\times(B\cap\Phi)=A\times\Phi \]
The Cartesian product of any set with the empty set is the empty set.
Hence,
\[ A\times\Phi=\Phi \]
Therefore,
\[ A\times(B\cap\Phi)=\Phi \]
So, the given statement is:
\[ \boxed{\text{True}} \]