Answer all questions on test. Use back if necessary. No scrap paper will be graded. (1:15) Exercise 1 Show by induction that: (I-x)Xl+x+x+x +-...xfor n 20 Exercise 2: Consider the following argument: If it is raining then the courts are closed The courts are open It is not raining Write this argument using propositional functions Is this argument valid? Justify your answer Exercise 3: Consider the set of strings on (0, 1such that L(A)wolw E (0,1 1) Show that the language is regular Exercise 4 Consider the following automaton: Give its formal definition. 2. 1. Which of the following strings is accepted by the automaton bb, a, E 3. Find an equivalent deterministic automaton Exercise S Consider the set X-(people in the world), and the relation R (xy) iff x and y have the same age) Is the relation reflexive, symmetric, anti-symmetric, transitive