$6\mathrm{.}4\mathrm{.}1$: Composition of relations.

$558630\mathrm{.}2917400.$qx$3$zqy$7$

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Given the following relations U and V$,$ use ordered pair notation to express the relation V$$U$.$

V$$U $=\{$

U

V

$\}$

Hint: Start with a pair of vertices such as $($a$,$ b$).$ If there exists some intermediate vertex that a can reach through one arrow in U and that intermediate vertex can reach b through one arrow in V$,$ then $($a$,$ b$)$ is in V$$U$.$ Otherwise, $($a$,$ b$)$ is not in V$$U$.$

$559630\mathrm{.}2017400$ ppariy $7$

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Given the following relations $\backslash ($ U $\backslash )$ and $\backslash ($ V $\backslash ),$ use ordered pair notation to express the relation $\backslash ($ V $\backslash$circ U $\backslash ).$

$\backslash [$

V $\backslash$circ U$=\backslash \{$

$\backslash ]$

Hint: Start with a pair of vertices such as $($a$,$ b$).$ If there exists some intermediate vertex that a can reach through one arrow in $\backslash ($ U $\backslash )$ and that intermediate vertex can reach $\backslash ($ b $\backslash )$ through one arrow in $\backslash ($ V $\backslash ),$ then $\backslash ((\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\})\backslash )$ is in $\backslash ($ V $\backslash$circ U $\backslash ).$ Otherwise, $\backslash ((\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\})\backslash )$ is not in $\backslash ($ V $\backslash$circ U $\backslash ).$