3. (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 $\neq$ 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) (4 points) A $k$-anticlique is a set of $k$ vertices, none of which share an edge with
each other. Write a statement in first order logic asserting that there is a 3-
anticlique in G.