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3) Prove the following logical equivalence WITHOUT using truth tables. You may use the rules from Section 2.9 and any of the extra rules presented in class. 4) Define the open statements P(A) and Q(A) as 7 P(A) AC (1,2,3) Q(A) : AS [2,3,4) Let S P({1, 2, 3, 4)). a) Determine all A e S such that P(A) Л~Q(A) is true. b) Determine all A e S such that P(A) A Q(A) is true. c) Determine all A E S such that P(A)Q(A) is true. 5) The following is a statement of a famous conjecture in number theory. It was proved a couple of years ago. There exists a natural number k such that for all natural numbers N, there exists primes p and q such that p > N, q> N, and lp- ql < k. a) Using the open statements P(x): is prime, Qz, y):> y, and R(x, y, 2): a -yl <z, write this statement in symbols. b) Write the negation of the statement in symbols. c) Write the negation of the statement in wor

number 5 please

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