Define p = graph G is bipartite (a) Represent the following statements symbolically using the statement vari- ables given above. Do not simplify your answers. i. Graph G is bipartite if and only if it has only cycles of even length. q = graph G has only cycles of even length ii. Graph G being bipartite is a necessary condition for G only having cycles of even length, and G being bipartitie is a sufficient condition for G only having cycles of even length. (b) Show that the statement given in part (i) is logically equivalent to the statement given in part (ii) using methods from class. (c) Negate the statement given in part (ii).

Can someone provide the proof for this? TIA

II. Sketch the graph of the inequality. 2.) 5x + 3y 2 - 15 y 4 61 4- 3- 21 192 ms ari 2i t92 noituloz erit 16rj st6ibni noisulce.oh -6-5-4 -3 -2 -1 1 2 3 4 5 6 -11 -21 -3 -4- 2. -5- -6-

Refer to the accompanying graph. Complete parts (a) through (c) in order. Starting at A, the circuit is Starting at B, the circuit is Starting at C, the circuit is Starting at D, the circuit is Starting at E, the circuit is (Simplify your answers.) (b) Which of the circuits for M A. The circuit that stai TL-- AL - and the total weight is otal weight is otal weight is otal weight is otal weight is A D C E→B→A A B C D→E→A A C D E→B→A ... A-E→C D→B→A B 13 15 60 C (a) Use the nearest neighbor algorithm starting at each of the vertices in turn to determine an approximate solution to the problem of finding a minimum Hamilton circuit for the graph. In each case, find the total weight of the circuit. 6/ 57 14 9 8 E 12 D ion to the problem of finding a minimum Hamilton circuit for the graph? Select all that apply.