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Problem(1) (10 points) Complete the following Truth table. P q p q pq PVq PA -9 -PV -9 -(pAq) -(PVq) T T T F F T F F P P 9 P q p q p - q p q q p q -p q -p T F F F Problem(2) (a) Prove that v3 is irrational. (b) Prove that In 3 is irrational. (c) Prove that the sum of an irrational number and a rational number is irrational using a proof by contradiction. (d) (5 points) Show that there exist irrational numbers r and y such that r' is rational. Problem(3) Use mathematical induction to prove the following: (a) (5 points) The harmonic numbers H,, j = 1, 2, 3, . . . are defined by H, = 1+ 3 Prove that Han 2 1 + 5 whenever n is a nonnegative integer. What will be the limit of limn- Ek-1 E? (b) (5 points) Prove that for n 2 2 and n E N 9 What will be the limit of lim,-. Ek=1 7? (c) (5 points) Prove that ER_1 23 = [2(n+1)]2 NUA Problem(4) (a) (5 points) Let m be a positive integer. If a = b (mod m) and c = d (mod m), then a + c = b+ d(modm) and ac =bd (mod m). (b) (5 points) Prove that if al b and bla , then a = b or a = -b. (c) (5 points) Prove that if n E N and n is odd, then n2 = 1 (mod 8). Problem(5) (5 points) Convert (345)10 to its binary, octal, and hexadecimal notations. Problem(6) (a) (5 points) Show that if 2" - 1 is prime, then n is prime. [Hint: Use the identity 2ab - 1 = (20 - 1) . (20(6-1) + 20(6-2) + ... 20 + 1).] (b) (2 points) Use the result in (a) to identify which of the following is (are) NOT prime? 213 215 - 217 - 221_