Discrete Mathematics and Its Applications with MathZone

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ICS.6D, Fall 2007, Quiz 1, October 8, 2007 1) Determine whether each of these statements is T rue or F alse, by circling the correct answer: If 5=5, then 10=10 T or F If 5=5, then 10=749 T or F If 5=0, then 10=10 T or F If 5=0, then 10=749 T or F 2) a) Write the truth table for the following proposition: ( p V q ) ¬ p . (Extra columns have been provided for your work. Please label those you use.) An s: p ( p V q ) → ¬ p b) Using the above table, decide if the following is a tautology , contradiction , or contingency : ( ( p V q ) ¬ p) ¬ p : ______________ 3) Determine the truth value of each statement below, assuming the domain is integers : 5 x (x > 6) _________________ 2200 x (x > 6) _________________ 2200 x 5 y (y > x) _______________ 5 x 2200 y (y > x) __________________ 4) Let L(x) = “x loves Larry”, and let J(x) = “x loves Jerry”. Express the following statements using quantifiers and predicates L(x) and J(x): a) “Some people love Larry and some people love Jerry.” : __________________ b) “Some people love Larry but don’t love Jerry.” : __________________ c) “Everyone who loves Larry must also love Jerry.” : __________________
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d) “If everyone loves Larry then everyone loves Jerry.” : __________________ ICS.6D: Discrete Mathematics For Computer Science, Fall 2007, Quiz 2 1) Label each of the following T rue or F alse: {1} {1} ____, {1} {1} ____, {1} {1} ____, [21pts] {1} ___, {1} ____, {1} ____, { } ____ 2) [21pts] If A = {s, t, t, q, r, t, q} and B = {t, v, t}, then |A| = ______, |P(A)| = __________, |A × B| = ________ A B = _________________________ A – B = ______________________________ A B = _________________________________________, {r,s,r} × B = ______________________________________ 3) [12pts ] Compute the following values: - 4.3 = _______ 4.3 = ________ 4/3 = _______ - 4/3 = ________ 4) [20pts] State whether each of these functions from Z to Z ( Z =Integers ) is I njective (one-to- one ), S urjective (onto ), B ijective, or N either, and briefly explain/justify each answer. f(n) = 5n+1 ___________________________________________________________ f(n) = n 2 - 4 ___________________________________________________________ f(n) = n / 5 ___________________________________________________________ f(n) = 5 - n ____________________________________________________________ 5) [26pts] Prove the following claim (for any sets A and B): A (B - A) Do it on the next page where we list several facts that can be useful in this claim. You don’t have to use all the facts listed or explain which of the stated facts you are using. We’ll give partial credit if some part of your argument is correct and other part is not. If you are not sure how to prove it using logical argument, you can argue it using Venn Diagram’s or Set
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Membership tables, and you’ll get [16pts] for either correct Venn diagram argument or correct Set Membership table argument . Claim: A (B - A) Your proof: As a reminder, here are several facts about sets and logic which can be useful in this proof: Fact 1: x A B iff x A ^ x B [definition of set intersection] Fact 2: x A – B iff x A ^ x B
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