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Oregon State University, Corvallis CS 225 Midterm Reviews. Q&A And worked Solutions.

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CS 225: Midterm Exam Solutions – Spring 2017 Question 1: 1. Determine the contrapositive of the following two statements: (a) It is necessary to have a valid password to log on to the server. (b) If x is nonnegative, ...

CS 225: Midterm Exam Solutions – Spring 2017 Question 1: 1. Determine the contrapositive of the following two statements: (a) It is necessary to have a valid password to log on to the server. (b) If x is nonnegative, then x is positive or x is 0. 2. Write the negation of each following statements : (a) Either you pay the bill or you lose the company's services. (b)A positive integer is a prime only if it has no divisors other than 1 and itself. Question 2: Let U be the set of all problems on a comprehensive list of problems in science. Define four predicates over U by - P (x): x is a mathematics problem Q(x): x is time consuming R(x): x is easy S(x): x is solvable Translate each of the following formulas into English sentences :1. ∀? ( ?(?) → (?(?) ∨ ~?(?)) 2. ~∀? (?(?) → ?(?)) 3. ∃? ( ?(?) ∧ ?(?)) 4. ~∃? (?(?) ∧ ?(?)) Question 3: Let B(x), W(x), and S(x) be the predicates B(x) : x is a female W(x) : x is a good athlete S(x) : x is young Express each of the following English sentences in terms of B(x), W(x), S(x), quantifiers, and logical connectives. Assume the domain is all people. ( You may need to use these symbols : : ≥ ≤ ≠ ¬ ∧ ∨ ⊕ ≡ → ↔ ∃ ∀ ) a) All young people are not good athletes. b) Some female are not good athletes but they are young. c) Not all young people are good athletes. d) There is someone who is neither a good athlete nor a female. Question 4: Use truth tables to prove or disprove that the two compound propositions (P →( Q ν R)) and ((P ∧ ¬R) → Q) are logically equivalent. Question 5: Use direct method to prove that given any two rational numbers r and s with r < s, there is another rational number between r and s Question 6: Proof by contradiction to show that if k is an integer and 5k + 4 is odd then k is odd. Question 7: Let A = {x ∈ Z| x = 6a + 4 for some integer a}, B = {y ∈ Z | y = 18b − 2 for some integer b} Prove or disprove each of the following statement - B ⊆ A Question 8: Construct an algebraic proof that for all sets A and B, = ( Bc U ( Bc - A)) c Cite a property from Theorem 6.2.2 for every step of the proof. Theorem 6.2.2.png Question 10: Use mathematical induction to prove that for any positive integer n, 30 + 31 + .................+ 3n = 1/2 (3n+1 - 1)

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