Consider graphs in which there is at most one edge between any two vertices. Use proof by induction to show that under this condition a graph with n vertices has at most n^2 edges. Use proof by contradiction to show that the set of all prime numbers is infinite. Find grammars for sigma = {a, b} that generate the sets of a) All strings with exactly two a's b) All strings with at least three a's. The reverse of a string can be defined by the recursive rules a^R = a: (wa)^R = aw^R: for all a elementof sigma, w elementof sigma*. Prove that a) (uv)^R = v^R u^R, for all u, v elementof sigma^+. b) (w^R)^R = w for al w elementof sigma. Show that a) The grammars S rightarrow aSb|ab|lambda and S rightarrow aaSbb|aSb|ab|lambda are equivalent. b) The grammars S rightarrow aSb|bSa|SS|a and S rightarrow aSb|bSa|a are not equivalent