CS 30
First Long Examination
Reviewer
p → (q ∧ r) ≡ (p → q) ∧ (∼ p ∨ r)
1) Use a truth table to prove that
2) Convert the statements into symbolic notations then test the validity of the argument using the rules of inference: If I study, then I will not fail mathematics. If I do not study then I will play computer games. I failed mathematics. ——————————————————————– Therefore I must have played computer games. 3) A Mersenne prime is a prime number which can be written as one less than a power of two. (a) Find three Mersenne primes, that is, primes of the form 2n − 1 for some positive integer n. (b) Prove by contraposition that if 2n − 1 is prime then n is prime. (Hint: x − y is a factor of xn − y n , for any positive integer n.)
(c) Prove that the converse of part (b) is false. (Hint: take n = 11.) (d) Prove by contradiction that if a and n are a positive integers such that an − 1 is prime, then either a = 2 or n = 1. (e) Prove that if 2n − 1 is a Mersenne prime, then 2n−1 (2n − 1) is a perfect number. Note: A perfect number is a positive integer that is equal to the sum of its proper positive divisors. (e.g. The proper positive divisors of 28 are 1, 2, 4, 7 and 14. 28 is a perfect number since 28 = 1 + 2 + 4 + 7 + 14.)
4) One particular algorithm used to sort a list of numbers into increasing order is heapsort. Heapsort employs a preprocessing step called heapify. It takes 0 · 2n + 1 · 2n−1 + 2 · 2n−2 + 3 · 2n−3 + · · · + (n − 1) · 21 + n · 20 steps to “heapify” a list of 2n numbers. Prove by mathematical induction that the closed form for this summation is 2n+1 − n − 2 for n ≥ 0. 5) Consider a 2n × 2n chessboard with one arbitrarily chosen square removed (an example of such a chessboard is shown at the right). Prove that any such chessboard can be completely covered without gaps or overlaps by L-shapes consisting of 3 squares each
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6) [Pre-condition: m is a nonnegative integer, x is a real number, i = 0, and exp = 1.] while i 6= m exp := exp · x i := i + 1 end while [Post-condition: exp = xm ] loop invariant: I(n) is “exp = xn and i = n”.
an example of a 23 × 23 chessboard with one square removed
To prove the correctness of a loop with respect to its pre- and post- conditions, we may show that the four properties in the loop invariant theorem are satisfied. What are these four properties? Prove that the given loop satisfies these properties.