Discrete Math Final Exam Questions With 100% Correct And Verified Answers.

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. 1. Let x and y be two odd integers. We shall show that x − y is even. 2. Since x is odd, then x = 2k + 1 for some integer k. Since y is odd, then y = 2j + 1 for some integer j. 3. Let x − y = (2k + 1) − (2j + 1). 4. 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)^2 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 hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption - correct answer direct proof proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true - correct answer proof by contrapositive N - correct answer natural numbers; all numbers greater than or equal to 0 Z - correct answer set of all integers; ...-2, -1, 0, 1, 2... Q - correct answer rational numbers; all real numbers that can be expressed as a/b R - correct answer real numbers If every element in A is also an element of B - correct answer subset If A ⊆ B and there is an element of B that is not an element of A - correct answer proper subset the set of all elements that are elements of both A and B; ∩ - correct answer intersection the set of all elements that are elements of A or B; ∪ - correct answer union Provided set A1 = {1, 3, 2, 5} and A2 = {2, 5, 4, 6}, A3 = {2, 3, 5, 7}, then ∪ 3 i=1Ai = (a) {1, 2, 3, 4, 5, 6, 7} (b) {1, 3, 2, 5, 2, 5, 4, 6, 2, 3, 5, 7} (c) {2, 5} (d) {{1, 3, 4, 6}, {4, 5, 6, 7}, {1, 7}} - correct answer a 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^2 (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