$(1)$ The following diagram encodes an order among choices in $B-=\{a,b,c,d,e\},$ we let $x>-y$ if

there is a directed path $($i$.$e$.$ a concatenation of directed edges$)$ joining $x$ and $y.$ Answer the

following:

Check that $$ is not transitive in this example.

In Exercise $3($from Exercises List Part I$)$ we concluded that $$ is transitive if $>-$ is a

preference order. Why $>-$ is then not a preference order?

Check that there is no representation $U$ of $>-($Hint: derive a contradiction from the

assumption of the existence of such $U).$