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Problem 2. 1. Let E be a binary relation symbol representing adjacency in graphs. (That is, E(1,4) means in a graph that "the vertices x and y are adjacent.) Write a formula (x, y) in the first-order logic over the language L = (E) with equality expressing that "r and y have exactly two common neighbors. Note that except logical symbols you may use only E and =. (The phrases "r and y are adjacent and r and y are neighbors have the same meaning.) 1 2. Find a model and a non-model of a theory T = {(V2) - E(x,x), (V.2)(Wy) (E(x, y) + E(y, 2)), (V2)(By) 4(x, y)} over the language L. By a non-model of T we mean a structure of the same language that is not a model of T. 3. Is the formula (5.c)(Vy)(,y) provable or refutable from T (in a sound and complete proof system using the axioms of T)? Give an explanation for your