Solution There are 36 elements in the sample space $S=\{(x, y): x, y=1,2,3,4,5,6\}$ Then

$A =\{(1,1),(1,3),(1,5),(2,2),$ $(2,4),(2,6),$ $(3,1),(3,3),$ $(3,5),(4,2),(4,4),$ $(4,6),(5,1),(5,3),(5,5),(6,2),$ $(6,4),(6,6)\} $

$B =\{(1,2),(2,1),(1,5),$ $(5,1),(3,3),$ $(2,4),(4,2),(3,6),$ $(6,3),(4,5),(5,4),(6,6)\}$

$C=\{(1,1),(2,1),(1,2)\}$ and $D=\{(6,6)\}$

We find that

$A \cap B=\{(1,5),(2,4),(3,3),(4,2),(5,1),(6,6)\} \neq \phi$

Therefore, $A$ and $B$ are not mutually exclusive events.

Similarly $A \cap C \neq \phi, $ $A \cap D \neq \phi, $ $B \cap C \neq \phi$ and $B \cap D \neq \phi$

Thus, the pairs of events, $(A, C),(A, D),(B, C),(B, D)$ are not mutually exclusive events.

Also $C \cap D =\phi$ and so $C$ and $D$ are mutually exclusive events.