Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurence of the other event.Hence the two events A and B cannot occur simultaneously.

Thus $P(A\cap B)=0$

$\therefore$ Given a die is rolled.

Event E-die shows 4={4}

Event F-die shows even number {2,4,6}

$\therefore E \cap F$=${4}\neq\phi$

$\therefore E \cap F\neq\phi,E$ and $F$ are not mutually exclusive.