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Discrete Math Final Exam
What is a proposition? - correct answer a statement that is either true or false A compound proposition with 5 propositional variables. The number of rows in its truth table is (a) 2 × 5 = 10 (b) 2^5 = 32 (c) 5^2 = 25 (d) None of the above - correct answer b The logic expression p → q means that p cannot be True when q is False. (a) True (b) False - correct answer a Select the converse of p → q (a) p → q (b) q → p (c) ¬p → ¬q (d) ¬q → ¬p - correct answer b Which statement is the contrapositive of: "If x = 4, then 3x = 12." (a) If x = 4 then 3x = 12. (b) If 3x = 12 then x = 4. (c) If x /= 4 then 3x /= 12. (d) If 3x /= 12 then x /= 4 - correct answer d Which of the followings does NOT have the same meaning of p → q? (a) q if p
(b) p is sufficient for q (c) p is necessary for q (d) ¬q → ¬p - correct answer a Identify which line has a mistake in the proof of the theorem: The difference between two odd numbers is even.
- Let x and y be two odd integers. We shall show that x − y is even.
- Since x is odd, then x = 2k + 1 for some integer k. Since y is odd, then y = 2j + 1 for some integer j.
- Let x − y = (2k + 1) − (2j + 1).
- Since x−y is two times an integer, then x−y is even. (a) Line 1 (b) Line 2 (c) Line 3 (d) Line 4 - correct answer c m = 8 is an even integer since 8 = 2 · 4. m^2 = 8^2 = 64 is an even integer since 64 = 2 · 32. Therefore if n is an even integer, then n2 is also an even integer. - correct answer "Generalizing from examples" If n is an odd integer, then n = 2k+1 for some integer k. Therefore n^2 = (2k+1)^ and n^2 is odd. - correct answer "Skipping steps" If n is an odd integer, then n = 2k+1 for some integer k. Let n^2 = 2j + 1 for some integer j. Since n^2 is equal to two times an integer plus 1, then n^2 is odd. - correct answer "Circular reasoning" Suppose r is a rational number. The product of any two rational numbers is rational. Therefore r^2 = r · r is also rational. - correct answer "Assuming facts that have not yet been proven"
the set of elements that are a member of exactly one of A and B, but not both; ⊕ - correct answer symmetric difference The law that establishes the set equality A ∩ /(B ∪ C)/ = A ∩ (/B ∩ /C) is (a) De Morgan's law (b) Associative law (c) Absorption law (d) Distributive law - correct answer a intersection is empty (A ∩ B = ∅) - correct answer disjoint a non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets - correct answer partition Let X = {x, y, z}. Is the string zzyzx an element in X^4? - correct answer no If f maps an element of the domain to zero elements or more than one element of the target - correct answer not well defined Which of the followings is a well-defined algebraic function from R to R? (a) f(x) = √ x^ (b) f(x) = √ x (c) f(x) = 1 / x^2 − 2 (d) None of the above - correct answer a ceiling ALWAYS rounds - correct answer up floor ALWAYS rounds - correct answer down A function f: X → Y is ___ or ___ if x1 ≠ x2 implies that f(x1) ≠ f(x2) - correct answer one-to-one (every Y has only one mapping)
if the range of f is equal to the target Y - correct answer onto (every Y is mapped to an X) A function is ___ if it is both one-to-one and onto - correct answer bijective Let f : A → X be a function, where A = {a, b, c, d}, and X = {1, 2, 3, 4}. Then, which of the followings about the function f = {(a, 3),(b, 2),(c, 4),(d, 3)} is correct? (a) It is one-to-one. (b) It is an onto function. (c) It is a bijection. (d) None of the above. - correct answer d Let f : A → X be a function, then, which of the followings is not correct? (a) If |A| = |X|, then f must be a bijection. (b) If |A| > |X|, then f cannot be one-to-one. (c) If |A| < |X|, then f cannot be onto. (d) If |A| < |X|, then f may or may not be one to one. - correct answer d Which of the followings is correct? (a) log5 k + log5 2 = log5 (k + 2) (b) log5 k − log5 2 = log5 (k − 2) (c) (log5 k)/(log5 2) = log5 (k/2) (d) log5 k 2 = 2 log5 k - correct answer d 0 * 1 - correct answer 0 (multiplication acts as AND) 0 + 1 - correct answer 1 (addition acts as OR) Which of the followings is not correct?
OR gate - correct answer adds literals inverter gate - correct answer negates literals Select the function that is NOT Ω(n^2 ) (a) 2^n (b) n + 17 log n (c) 6n log n + 3n^2 + 2 (d) 2n + n! - correct answer a (check this one) Select the function that is O(2^n) (a) 3^n (b) n 5 + 2n log n (c) n! (d) 2^n log n + n - correct answer c (check this one) Select the function that is Θ(n log n) (a) 23n log log n + 3n log n (b) 15n + 17 log n (c) 6n log n + n^1.1 + 2 (d) 2^n log n + n - correct answer c (check this one) Provided f(n) = 100n+ (log n)^2 and g(n) = n + log 10n. Then, which of the followings best describes f and g? (a) f = Θ(g) (b) f = Ω(g) (c) f = O(g) (d) None of the above - correct answer c (check this one) n div d produces the: - correct answer quotient
n mod d produces the: - correct answer remainder -11 mod 4 - correct answer 1 17 mod 6 - correct answer 5 −344 mod 5 - correct answer 1 7 | 50 - correct answer false 8 | 40 - correct answer true 8 ∤ 79 - correct answer true What is the value of 5 times 7 in Z9? - correct answer 8 What is the value of 6 + 8 in Z4? - correct answer 2 x is congruent to y mod m if x mod m = y mod m - correct answer congruence relation Which number is congruent to 5639 mod 13? (a) 5621 (b) 5627 (c) 5653 (d) 5652 - correct answer d What is the value of ((131)^39 + 11 · (−11)) mod 13? (a) 3
A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. Each party must elect a chair and a vice chair from their party's members, and one person cannot be elected for both. How many different outcomes are there for the chair and vice chair elections? (a) 100 · 99 · 98 · 97 (b) 44 · 43 · 56 · 55 (c) 100^ (d) 44^2 · 56^2 - correct answer b a sequence of r items with no repetitions, all taken from the same set - correct answer r-permutation A license plate has 7 characters. Each character can be a capital letter or a digit except for 0. How many license plates are there in which no character appears more than once and the first character is a digit? (a) 9 · P(35, 6) (b) 9 · P(34, 6) (c) 9 · (35)^6 ) (d) 9 · (34)^6 ) - correct answer a There is a set of 10 jobs in the printer queue. Two of the jobs in the queue are called job A and job B. How many ways are there for the jobs to be ordered in the queue so that job A completes some time before job B? (a) 10!/ (b) 2 · 9! (c) 9! (d) 10! - correct answer b another word for subset, usually for counting situations in which there in no particular ORDER imposed on a set of outcomes - correct answer r-subset curly brackets indicate that: - correct answer the order of the elements does not matter (non-expanded permutations)
normal parenthesis indicate that: - correct answer the order of the elements does matter (usually expanded) to find r-subsets from set S ("n choose r"): - correct answer n! / r!(n-r)! A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. How many ways are there to pick a 10 member committee of senators with the same number of Reds as Blues? (a) 44 10 · 56 10 (b) 100 10 (c) 44 5 + 56 5 (d) 44 5 · 56 5 - correct answer d Natasha is in a class of 30 students that selects 4 leaders. How many ways are there to select the 4 leaders so that Natasha is one of the leaders? (a) 30 4 (b) 29 4 (c) 30 3 (d) 29 3 - correct answer d A state's license plate has 7 characters. Each character can be a capital letter (A-Z), or a digit except for 0 (1-9). How many license plates are there in which exactly 3 of the 7 characters are digits? (a) 7 3 · (35)^ (b) P(7, 3) · (35)^ (c) 7 3 · (26)^ (d) 7 3 · 9^3 · (26)^4 - correct answer a a way of ordering n-tuples in which two n-tuples are compared according to the first entry where they differ, goes smallest to largest
In an inductive proof of the theorem, what must be proven in the base case? - correct answer Q(1) is true In an inductive proof of the theorem, what must be proven in the inductive step? - correct answer For all positive integers k, Q(k) implies Q(k + 1) One mixed his steps in proving n! > 2^n for n ≥ 4 with mathematical induction. The right order of his steps numbered below should be:
- Assume k! > 2^k for k ≥ 4.
- Therefore, for any n ≥ 4, n! ≥ 2^n.
- When n = k+1, n! = (k+1)! = k!(k+1) > 2^k (k + 1) > 2^k+1. The last step is because of k + 1 > 2.
- When n = 4, n! = 4! = 24 and 2^n = 24 = 16. Hence, n! > 2^n for n = 4. (a) 3, 4, 2, 1 (b) 4, 3, 1, 2 (c) 2, 1, 3, 4 (d) 4, 1, 3, 2 - correct answer d
