Let G be a simple undirected graph with a set of vertices V. Let V₁. and V₂ be subsets of V so that V₁ UV₂ = Vand VinV₂ = 0. Let E(r, y) be the predicate representing that there is an edge from rz to y. Note that the graph being undirected means that Vu € V Vr € V (E(u, v) → E(v.u)).
(a) (6 pts) Express each of the following properties in predicate logic. You can only use V.V₁, V₂, E(-.-), logical and mathematical operators.
(i) Every edge connects a vertex in Vi and a vertex in V₂
(ii) For every vertex in V, there are edges that connect it with all vertices in V
(b) (2 pts) If (a)(i) is true, is G necessarily a bipartite graph? Please give brief justification.
(c) (2 pts) If (a)(ii) is true, is G necessarily a complete bipartite graph? Please give a brief justification.