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To view a copy of this license, visit  \n[http://creativecommons.org/licenses/by/4.0/ or](http://creativecommons.org/licenses/by/4.0/ or) send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.  \nAcknowledgements  \nThis work was made possible by an Open Educational Resource grant from the Division of Learning and Teaching Support and Innovation at the University of Victoria. Ideas for questions were taken from:  \n􀀏 Kieka Mynhardt's notes, assignments, and tests for Math 222  \n􀀏 Introduction to Combinatorics and Graph Theory - Custom Edition for the University of Victoria  \n􀀏 Discrete Mathematics: Study Guide for MAT212-S-Dr. Kieka Myndardt  \n􀀏 Discrete Mathematics-Norman L. Biggs  \n􀀏 Applied Combinatorics, fourth edition-Alan Tucker  \n􀀏 Discrete Mathematics, An Introduction to Mathematical Reasoning-Susanna  \nS. Epp  \n􀀏 Discrete Mathematics with Combinatorics-James A. Anderson  \nContents  \n1 Preliminaries 1  \n1.1 Sets .................................... 1  \n1.2 Relations and Graphs ........................... 7  \n2 Graph Theory 15  \n2.1 Graphing Preliminaries .......................... 15  \n2.2 De􀀌nitions and Basic Properties ..................... 15  \n2.3 Isomorphisms ............................... 29  \n2.4 Eulerian Circuits ............................. 44  \n2.5 Hamiltonian Cycles ............................ 56  \n2.6 Trees and Their Properties ........................ 71  \n2.7 Planar Graphs ............................... 82  \n2.8 Colouring Graphs ............................. 95  \n3 Counting: Fundamental Topics 111  \n3.1 Basic Counting Principles ........................ 111  \n3.2 The Rules of Sum and Product ..................... 111  \n3.3 Permutations ............................... 123  \n3.4 Combinations and the Binomial Theorem ................ 134  \n3.5 Combinations with Repetitions ..................... 149  \n3.6 The Pigeonhole Principle ......................... 162  \n4 Inclusion and Exclusion 172  \n4.1 The Principle of Inclusion-Exclusion ................... 172  \n4.2 Derangements: Nothing in its Right Place ............... 194  \n4.3 Onto Functions and Stirling Numbers of the Second Kind ....... 203  \n5 Generating Functions 213  \n5.1 Introductory Examples .......................... 213  \n5.2 De􀀌nition and Examples: Calculating Techniques ........... 213  \n5.3 Partitions of Integers ........................... 236  \n6 Recurrence Relations 247  \n6.1 First-Order Linear Recurrence Relations ................ 247  \n6.2 Second Order Linear Homogeneous Recurrence Relations with Constant Coe􀀎cients ............................. 254  \n1 Preliminaries  \n1.1 Sets Questions:  \n1. Consider the sets A and B where:  \nA = fa 2 Z j a = 2k, for some integer kg,  \nB = fb 2 Z j b = 2j 􀀀 2, for some integer jg.  \nDoes A = B? If yes, prove it. If no, explain why not.  \n2. Consider the sets A = f1 ; 2 ; 3g, B = fx; yg, and C = fu; vg. Let P (A) denote the powerset of A. Find each of the following:  \n(a) P (A [ B)  \n(b) P (B 􀀂 C)  \n(c) P (P(C))  \n(d) A 􀀂 (B \\ C)  \n(e) (A 􀀂 B) 􀀂 C  \n3. Prove the following: The empty set is a subset of every set.  \n4. Prove or disprove: For the arbitrary sets A; B; and C, knowing that A 􀀒 B and A 􀀒 C implies that A 􀀒 B \\ C.  \n5. Prove the following identity:  \nA 􀀂 (B [ C) = (A 􀀂 B) [(A 􀀂 C)  \nfor arbitrary, nonempty sets, A; B and C.  \n6. Prove that for any three arbitrary sets, A, B, and C, if C 􀀒 B 􀀀 A, then A \\ C = ; .  \n7. If the statement about powersets is true, prove it, or 􀀌nd a counterexample if it is false. For all sets A and B ,  \n(a) P (A [ B) 􀀒 P (A) [ P (B)  \n(b) P (A \\ B) = P(A) \\ P (B)  \n(c) If A 􀀒 B then P(A) 􀀒 P (B)  \n8. Prove this statement of De Morgan's Laws:  \nA \\ B = A [ B:  \nSolutions:  \n1. Yes, A = B . We will p","cbCaipdyDtkSaFBQ","https://ap.wps.com/l/cbCaipdyDtkSaFBQ","pdf",855636,280,"English","# Preliminaries\n## Sets\n## Relations and Graphs\n# Graph Theory\n## Graphing Preliminaries\n## De nitions and Basic Properties\n## Isomorphisms\n## Eulerian Circuits\n## Hamiltonian Cycles\n## Trees and Their Properties\n## Planar Graphs\n## Colouring Graphs\n# Counting: Fundamental Topics\n## Basic Counting Principles\n## The Rules of Sum and Product\n## Permutations\n## Combinations and the Binomial Theorem\n## Combinations with Repetitions\n## The Pigeonhole Principle\n# Inclusion and Exclusion\n## The Principle of Inclusion-Exclusion\n## Derangements\n## Onto Functions and Stirling Numbers of the Second Kind\n# Generating Functions\n## Introductory Examples\n## De nition and Examples: Calculating Techniques\n## Partitions of Integers\n# Recurrence Relations\n## First-Order Linear Recurrence Relations\n## Second Order Linear Homogeneous Recurrence Relations with Constant Coe fficients","[{\"question\":\"What topics are covered in the practice questions?\",\"answer\":\"The document covers sets, relations, graphs, and graph theory topics such as isomorphisms, Eulerian circuits, Hamiltonian cycles, trees, planar graphs, and colouring. It also includes counting, inclusion-exclusion, derangements, onto functions, generating functions, partitions of integers, and recurrence relations.\"},{\"question\":\"Are solutions included for each practice question?\",\"answer\":\"Yes. The document is explicitly labeled as “Practice Questions with Solutions,” and the provided section shows questions followed by worked solutions.\"},{\"question\":\"How are key concepts in sets typically tested in the questions?\",\"answer\":\"The questions use subset and powerset reasoning, set operations like union, intersection, complement, and Cartesian product, and they ask for proofs or counterexamples. For example, problems compare sets for equality and apply De Morgan’s Laws.\"}]","MATH 222 Discrete and Combinatorial Mathematics - Practice Questions with Solutions | PDF",1789653808,706]