Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Material Type: Exam; Class: Introduction to Discrete Mathematics; Subject: MATHEMATICS; University: University of Wisconsin - Madison; Term: Fall 2005;

Typology: Exams

Pre 2010

1 / 4

Download Circle java, Rectangle java, Shapes Tester java - Lab 9 | MATH 240 and more Exams Discrete Mathematics in PDF only on Docsity! MATH 240; EXAM # 1, 100 points, October 18, 2005 (R.A.Brualdi) TOTAL SCORE (11 problems; 100 points possible): Name: These R. Solutions TA: Anders Hendrickson (circle time) Mon 12:05 Mon 1:20 Wed 12:05 Wed. 1:20 1. [8 points] Let P (x) and Q(x) be predicates where the universe of discourse for x is some set U . Let A = {x : P (x) is true} and let B = {x : Q(x) is true} be the truth sets of P (x) and Q(x), respectively. Circle all the predicates below that have truth set equal to A ∩ B? (a) YES P (x) ∧ ¬Q(x) (b) YES ¬(P (x) → Q(x)) (c) YES ¬(¬P (x) ∨ Q(x)) (d) ¬(Q(x) ∨ P (x)) (e) ¬(P (x) ∧ Q(x)) 2. [8 points] Let f : A → B and g : B → C be functions where A, B, C are finite sets. Circle all the CORRECT statements below. (a) If f is surjective, then |A| ≤ |B|. (b) If |A| ≤ |B|, then f is injective. (c) YES If f is surjective and |A| = |B|, then f is injective (d) YES If f and g are both injective, then g ◦ f is injective. (e) If g is surjective, then g ◦ f is surjective. 1 3. [8 points] Circle all the CORRECT statements below, or circle (e): (a) YES d−ne = −bnc. (b) d−2.999999999e = −3 (c) dx + ye = dxe + dye (d) YES dx − 0.5e is the closest integer to x, rounding down in the case of ties. (e) None are correct. 4. [6 points] The number of different functions f : A → B from a set A of m elements to a set B of n elements equals: (a) mn (b) YES nm (c) mn (d) m + n (e) None of the above 5. [10 points] For each of the functions f(n) below, give the simplest function g(n) such that f(n) = Θ(g(n)). (a) .01n3 − 1000n2 + 5n + 35: Θ(n3) (b) 4n 5+3n4 log n−3n+5 2n3+5n2−6n+8 : Θ(n2) (c) f(n) = bncn: Θ(n2) (d) f(n) = sin n: Θ(1) 2