Quiz 1-4 Solutions

Discrete Mathematics and Its Applications with MathZone

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Quiz 1: (all questions viable for midterm!) 1) Determine whether each of these statements is T rue or F alse, by circling the correct answer: [6 pts each] If 5=5, then 10=10 T If 5=5, then 10=749 F If 5=0, then 10=10 T If 5=0, then 10=749 T 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.) [3 columns to write, 12 pts] Ans : p ( p V q ) →¬ p F F T T b) Using the above table, decide if the following is a tautology , contradiction , or contingency : ( ( p V q ) ¬ p) ¬ p [8 pts] tautology, because the corresponding columns are same 3) Determine the truth value of each statement below, assuming the domain is integers : [6 pts for each] 5 x (x > 6) ____T _____________ 2200 x (x > 6) ___ F ______________ 2200 x 5 y (y > x) __ T _____________ 5 x 2200 y (y > x) ____ F _____________ 4) Let L(x) = “x loves Larry”, and let J(x) = “x loves Jerry”. [6 pts for each] Express the following statements using quantifiers and predicates L(x) and J(x): a) “Some people love Larry and some people love Jerry.” : 5 x L(x) ^ 5 x J(x) b) “Some people love Larry but don’t love Jerry.” : 5 x ( L(x) ^ ¬ J(x) ) c) “Everyone who loves Larry must also love Jerry.” : 2200 x ( L(x) => J(x) )

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d) “If everyone loves Larry then everyone loves Jerry.” : 2200 x L(x) => 2200 x J(x) [last two were pretty tricky…] Quiz 2: (all questions viable for midterm!) 1) Label each of the following T rue or F alse: [18pts] {1} {1} _T (case of A A), {1} {1} _F (not true because {1}={1}), {1} {1} _ F {1} __F , {1} _T_( A for all non-empty sets A) { } _T__ 2) [21pts] If A = {s, t, t, q, r, t, q} and B = {t, v, t}, then. ( First note that A={s,t,q,r} and B={t,v}) |A| = __ 4 ____, |P(A)| = ___ 2 4 =16 _______, |A × B| = __ 4*2=8 ______ A B = ___ _{t} A – B = ______{s,q,r} A B = _____ {s,t,q,r,v} {r,s,r} × B = ____ {r,s} x B = {(r,t),(r,v),(s,t),(s,v)} 3) [12pts ] Compute the following values: - 4.3 = __ -5 _____ 4.3 = ___ 5 _____ 4/3 = __ 1 _____ - 4/3 = ___ -1 _____ 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.
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