$(16$ points$)$ In this part we state graph theory properties in first order logic. Let $G=(V,E)$ and let the predicate $E(x,y)$ mean that there is an $($undirected$)$ edge between $x$ and $y.$ In your answers, all quantifiers should have universe $V,$ the vertices of the graph.

$($a$)(4$ points$)$ Write a statement in first$-$order logic which asserts that for every pair of distinct vertices $x$ and $y,$ there is a path from vertex $x$ to vertex $y$ of length exactly $3.$ That is$,$ there should be exactly $2$ vertices in between. $($That means that each of those two vertices have to be different from each other and different from $x$ and $y.$ This may require using $$ and several conjunctions.$).$

$($b$)(4$ points$)$ Write a statement in first$-$order logic which asserts that there is a vertex $x$ and a cycle of length $3$ starting and ending at vertex $x.$

$($c$)(4$ points$)$ Write a statement in first$-$order logic asserting that there is a vertex $x$ with degree at least $2.$

$($d$)$ points