1. Given the state-transition graph (Kripke Structure) below, which of these properties hold • (AX f1)(s0) (EG f2)(s2) • A [b1 U f1] (s3) AG (b1 = AF f1) • AG (f1 V f2)

4. (1) Draw a transition graph for the dfa M={Q,E,8,q,,F ), where Q={q,,91»92 }; E = {a,b},F = {qo,92} and ở is definded as S(90,a) = q,,8(q,b) =q,,5(q,,a)=q0,8(q,,b) =q2,8(q2,a) =q2,8(q2,b) =q2 (2) Give the language accepted by the above dfa.

Consider the following labelled transition system. a a b Determine whether the states s and t are strongly bisimilar. (Justify your answer!)

Question 1 Refer to the following network flow and state the min cut, of the graph G. S 3/7 6/6 -5/5 (s,v1),(s,v3),(s,v2) 1/1 (v1,v4), (v1,v3),(s,v3),(s,v2) O (v4,t).(v3,t).(v5,t) O (s,v2),(v2,v3),(v2,v5) 0/3 -2/2- 4/5 -2/3 6/6 6/8

5. Let f:X→Y and g:Y→Z be one-to-one correspondences. (a) Prove that g◦fmust be a one-to-one correspondence. (b) Prove that (g◦f)−1=f−1◦g−1.

C. The base function f(x) is reflected in the x-axis, stretched elly by a and translated 3 units to the left and 1 unit down Witte the equation of the function g(x) (4 marks). Given the graph of f(x), graph each of the following clearly labeling each graph: (2 marks each) a. f(-x) f-¹(x) ter of 2. fored b.