Final-Exam of International College, Semester 2 of 2025-2025 Discrete Mathematics Test form: Open-book exam Teachers: David Paper created by: David For Grade 2025 Undergraduate (Bachelor) Whether scratch paper needed: N (√ )/Y ( )Major: ____________Class: ____ Name: ________Student No.: _________ Class No.: ____PartsIIIIIITotalSignatureScoreNotes: 1 Answers need to be written on the exam papers. 2 Students are not allowed to take away exam papers out of the exam room.Part I- True or False (20 points, 1 points for each) 1.If T is a tree with 17 vertices, then there is a simple path in T of length 17.2.There is a tree with degrees 3, 3, 2, 2, 1, 1, 1, 1.3.If two trees have the same number of vertices and the same degrees, then the two trees are isomorphic.4.Every tree is planar.5.1 + 1 = 3 if and only if 2 + 2 = 3. 6.If 2 + 1 = 3, then 2 = 3 − 1. In the following questions suppose A = {x, y} and B = {x, {x}}. 7 8 ScoreSignature In the following questions suppose A = {a, b, c}. 9 10 In the following questions suppose A = {1, 2, 3, 4, 5}.11 12 13 In the following questions, suppose A = {a, b, c} and B = {b, {c}}. 14 15 16 17 Assume that the statement applies to all sets.18 19 20 Part II - Short answer questions (40 points, 2 points for each)What is the negation of the propositions in the following?21 Abby has more than 300 friends on facebook.22. Prove that (q (p –>¬q)) –> ¬p is a tautology using propositional equivalence and the laws of logic.23. Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if 3n + 5 is even, then n is odd. Which type of proof would be easier to give? Explain why.24. For each of...
