Let R$1$ be a relation from X to Y and R$2$ be a relation from Y to Z$.$

The composition R$2$ R$1$ is the relation from X to Z given by

R$2$ R$1=$ f$($x; z$)2$ X Z : for some y $2$ Y; $($x; y$)2$ R$1$ and $($y; z$)2$ R$2$g:

Then we have:

Theorem $1.$ Choose orderings of X$,$ Y and Z$.$ Let M$1$ be the adjacency matrix

of R$1$ and M$2$ be the adjacency matrix of R$2$ with respect to these orderings. Then

the adjacency matrix of the relation R$2$R$1$ is obtained by replacing each non$-$zero

entry in M$1$M$2$ by the number $1.$

Let X $=$ fa; b; cg$,$ Y $=$ f$1$; $2$; $3$; $4$g and Z $=$ f; ; $$g$.$ Let

R$1=$ f$($a; $1)$; $($a; $2)$; $($b; $3)$; $($b; $4)$; $($c; $1)$; $($c; $4)$g X Y

R$2=$ f$(1$; $)$; $(2$; $)$; $(2$; $)$; $(3$; $)$; $(4$; $)$g Y Z:

$($a$)$ Compute the adjacency matrices M$1$ associated with R$1$ and M$2$ associated

with R$2.$

$($b$)$ Use the theorem above to compute the adjacency matrix for the relation

R$2$ R$1.$

$($c$)$ Use your answer to part $($b$)$ to write R$2$ R$1$ as a subset of X Z$.$

$($d$)$ Write down the adjacency matrix M$3$ associated with $($R$2$ R$1)1$