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CompSci 230 Midterm Exam Sample— Sample Solution 1. Place T for “true” and F for “false” next to each of the statements below. P is the set of positive integers. 3|6 T 6|3 F 2 3=6 F If n ∈ P then n|0 T If a, b > 0 then log(a) + log(b) = log(ab) T ∅∈∅ F 15 and 18 are coprime F 2. Write numerical values for the following expressions in the table below. Each answer is a single number. 3. What is the prime factorization of 120? 7 mod 3 1 3 mod 7 3 GCD(2 × 33 × 42 , 22 × 3 × 4) 24 Answer: 23 × 3 × 5 4. What is the decimal representation of 1101{2}? 5. What is the binary representation of 26{10}? Answer: 13 Answer: 11010 6. Is this function tail recursive? (define (f n k) (cond ((> k n) 0) ((or (= k 0) (= k n)) 1) (else (+ (f (- n 1) (- k 1)) (f (- n 1) k))))) Answer: No. 7. What does the function in the previous question compute? Answer: (n) k 8. What does this Racket expression evaluate to? (map (lambda (x) (apply + (cons x '(2 3)))) '(1)) Answer: '(6) 9. Apply the substitution model to the following expression: (cond ((< 0 3) (- 5 (* 2 2))) (else 2)) Answer: (cond (#t (- 5 (* 2 2))) (else 2)) (- 5 (* 2 2)) (- 5 4) 1 COMPSCI 230 — Duke — February 22, 2015 10. Write the simplest possible expression for the set N \ P. Assume that zero is a natural number. Answer: {0} 11. Let A = {1, 2} and B = {2, 3, 4}. Write the following set by enumerating its elements explicitly. C = (B \ A) × (A ∩ B) Answer: {(3, 2) , (4, 2)} 12. Consider all 3-sequences from domain D = {1, 2, 3} into codomain C = {a, b, c}. Answer: 33 = 27. (a) How many such sequences are there? Answer: 3! = 6 (b) How many of such sequences are surjections? Answer: 3! = 6 (c) How many of such sequences are injections? 13. This is a row of Pascal’s triangle: 1 6 15 20 15 6 1 Write the subsequent row. Answer: 1 7 21 35 35 21 7 1 14. Let D = {x : x ∈ Z and |x| = x2 } and C = {y : y ∈ [2..5] and y | 6} (a) Write D by listing its elements explicitly. Answer: D = {−1, 0, 1} (b) Write C by listing its elements explicitly. Answer: C = {2, 3} (c) Let f be the function with domain D and codomain C defined as follows: f = {(−1, 3), (0, 2), (1, 2)} Is f an injection? Answer: No. (d) Is f a surjection? Answer: Yes. (e) How many functions are there from D to C? Answer: 23 = 8 (f) How many functions are there from C to D? Answer: 32 = 9 15. The following is an inefficient implementation for the RSA encryption of a cleartext message m with public key (n, e): (define (encrypt m e n) (modulo (expt m e) n)) Write the definition of an equally inefficient function decrypt that decrypts the resulting cypher text c with private key (n, d). Answer: (define (decrypt c d n) (modulo (expt c d) n)) 16. If A is a denumerable set (that is, it has the same cardinality as the set of natural numbers), is A × A always denumerable? Answer: Yes. 17. Should you use dovetailing (Cantor’s first diagonal argument) or diagonalization (Cantor’s second diagonal argument) to prove your previous answer? Answer: Dovetailing. COMPSCI 230 — Duke — February 22, 2015 18. Use the binomial theorem to prove that n ∑ ( ) n (−1) =0 k k k=0 Answer: Letting a = −1 and b = 1 in (a + b)n = n ( ) ∑ n k n−k a b k k=0 yields the result immediately. 19. There are three men and six women. Each man marries one of the women. In how many ways can this be done? Answer: Every 3-permutation of a 6-set corresponds to a different set of marriages, so the count is (6)3 = 6! = 6 · 5 · 4 = 120 . (6 − 3)! 20. Prove the following equality with a combinatorial argument: ( ) ( ) n n = i n−i for 0≤i≤n (no credit for an algebraic proof). ( ) Answer: There are ni ways( to pick ) i elements out of an n-set. Each choice is equivalent to identifying the n − i elements that n are not chosen, and there are n−i ways to do so. So the lefthand side and the righthand side of the equality above must be equal to each other. COMPSCI 230 — Duke — February 22, 2015