(2) [6 pts] In this problem we consider the language of (undirected, irreflexive) graphs. This language has only one binary relation symbol R. The axioms are: Vx. 7xRx and Vxy.xRy + yRx. When (G, R) is a graph, we refer to the elements of G as the vertices of the graph. (a) A path in a graph (G, R) is a sequence ao, ..., An of vertices such that a;Rai+1 for all i = 0,...,n 1. In this case we say that the length of the path is n. Write down a formula On(x,y) (in the language of graphs) with two free variables x, y, such that (G, R) = n(a,b) precisely when there is a path of length n from a to b. (You only have to give the formula, you don't have to prove that it does the right thing.) (b) A graph (G, R) is called n-connected when for every pair of distinct vertices a,b there is a path from a to b of length at most n. Show that the class of n-connected graphs can be axiomatized. (c) A graph (G, R) is called connected when for every pair of distinct vertices a, b there is a path from a to b. Show that the class of connected graphs is not axiomatizable. Hint: use compactness, and add to the supposed set of axioms two new constants a, b and for each n e N an axiom that expresses that there is no path of length