Prove that if n is an integer, then n2 3n + 5 is odd.
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1
Logic
1.1
Statements
1.2
Negation, Conjunction and Disjunction
1.3
Implications
1.4
Biconditionals
1.5
Tautologies and Contradictions
1.6
Some Applications of Logic
2
Sets
2.1
Sets and Subsets
2.2
Set Operations and Their Properties
2.3
Cartesian Products of Sets
2.4
Partitions
3
Methods of Proof
3.1
Quantified Statements
3.2
Direct Proof
3.3
Proof by Contrapositive
3.4
Proof by Cases
3.5
Counterexamples
3.6
Existence Proofs
3.7
Proof by Contradiction
4
MATHEMATICAL INDUCTION
4.1
The Principle of Mathematical Induction
4.2
Additional Examples of Induction Proofs
4.3
Sequences
4.4
The Strong Principle of Mathematical Induction
5
RELATIONS AND FUNCTIONS
5.1
Relations
5.2
Equivalence Relations
5.3
Functions
5.4
Bijective Functions
5.5
Cardinalities of Sets
6
ALGORITHMS AND COMPLEXITY
6.1
What is an Algorithm?
6.2
Growth of Functions
6.3
Analysis of Algorithms
6.4
Searching and Sorting
7
INTEGERS
7.1
Divisibility Properties
7.2
Primes
7.3
The Division Algorithm
7.4
Congruence
7.5
Introduction to Cryptography
7.6
Greatest Common Divisors
7.7
Integer Representations
8
INTRODUCTION TO COUNTING
8.1
The Multiplication and Addition Principles
8.2
The Principle of Inclusion-Exclusion
8.3
The Pigeonhole Principle
8.4
Permutations and Combinations
8.5
Applications of Permutations and Combinations
9
ADVANCED COUNTING METHODS
9.1
The Pascal Triangle and the Binomial Theorem
9.2
Permutations and Combinations with Repetition
9.3
Generating Functions
10
DISCRETE PROBABILITY
10.1
The Probability of an Event
10.2
Conditional Probability and Independent Events
10.3
Random Variables and Expected Values
11
PARTIALLY ORDERED SETS AND BOOLEAN ALGEBRAS
11.1
Partially Ordered Sets
11.2
Lattices
11.3
Boolean Algebras
12
INTRODUCTION TO GRAPHS
12.1
Fundamental Concepts of Graph Theory
12.2
Connected Graphs
12.3
Eulerian Graphs
12.4
Hamiltonian Graphs
12.5
Weighted Graphs
13
TREES
13.1
Fundamental Properties of Trees
13.2
Rooted and Spanning Trees
13.3
The Minimum Spanning Tree Problem
14
PLANAR GRAPHS AND GRAPH COLORINGS
14.1
Planar Graphs
14.2
Coloring Graphs
15
DIRECTED GRAPHS
15.1
Fundamental Concepts of Digraph Theory
15.2
Tournaments
15.3
Finite-State Machines
Textbook Solutions for Discrete Mathematics
Chapter 3.4 Problem 12
Question
In the proof of Result 3.28 it was proved, for integers m and n, that if mn is odd, then m and n are both odd. This was done using a proof by contrapositive. In the proof that was given, it was assumed, without loss of generality, that m is even. Give a proof of this implication, using a proof by cases namely: Case 1. m is even . Case 2. n is even.
Solution
The first step in solving 3.4 problem number 12 trying to solve the problem we have to refer to the textbook question: In the proof of Result 3.28 it was proved, for integers m and n, that if mn is odd, then m and n are both odd. This was done using a proof by contrapositive. In the proof that was given, it was assumed, without loss of generality, that m is even. Give a proof of this implication, using a proof by cases namely: Case 1. m is even . Case 2. n is even.
From the textbook chapter Proof by Cases you will find a few key concepts needed to solve this.
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Title
Discrete Mathematics 1
Author
Gary Chartrand, Ping Zhang
ISBN
9781577667308
In the proof of Result 3.28 it was proved, for integers m and n, that if mn is odd, then
Chapter 3.4 textbook questions
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Chapter 3: Problem 1 Discrete Mathematics 1
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Chapter 3: Problem 2 Discrete Mathematics 1
Prove that if n is an integer, then n3 n is even.
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Chapter 3: Problem 3 Discrete Mathematics 1
Let n Z. Prove that if n2 + n = 0, then 2n+3n 12n is even.
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Chapter 3: Problem 4 Discrete Mathematics 1
Prove that if n is an integer, then 3n + 1 and 5n + 2 are of opposite parity.
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Chapter 3: Problem 5 Discrete Mathematics 1
(a) Let m and n be two integers. Prove that m+ n is even if and only if m and n are of the same parity. (b) Let S = {a, b, c, d} be a set of four integers. Prove that if m pairs of distinct integers of S are of the same parity and the remaining n pairs of distinct integers of S are of opposite parity, then m and n are of the same parity.
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Chapter 3: Problem 6 Discrete Mathematics 1
Let m and n be two integers. Prove that 3mn is even if and only if m and n are of the same parity.
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Chapter 3: Problem 7 Discrete Mathematics 1
Let m and n be two integers. Prove that 7m+3n is odd if and only if m and n are of opposite parity.
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Chapter 3: Problem 8 Discrete Mathematics 1
Let m and n be two integers. Prove that if m and n are of opposite parity, then 2m + n and 3m 4n are of opposite parity.
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Chapter 3: Problem 9 Discrete Mathematics 1
Let m and n be two integers. Prove that mn2 is odd if and only if m and n are odd.
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Chapter 3: Problem 10 Discrete Mathematics 1
Let m and n be two integers. Prove that mn and m+ n are both even if and only if m and n are both even.
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Chapter 3: Problem 11 Discrete Mathematics 1
Give a proof of Let n Z. If |2n 1| 5, then n 3 and n 2. using (a) a direct proof. (b) a proof by contrapositive.
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Chapter 3: Problem 12 Discrete Mathematics 1
In the proof of Result 3.28 it was proved, for integers m and n, that if mn is odd, then m and n are both odd. This was done using a proof by contrapositive. In the proof that was given, it was assumed, without loss of generality, that m is even. Give a proof of this implication, using a proof by cases namely: Case 1. m is even . Case 2. n is even.
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Chapter 3: Problem 13 Discrete Mathematics 1
Give a proof of Result 3.29 where without loss of generality is used twice.
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Chapter 3: Problem 14 Discrete Mathematics 1
Use a proof by cases (as in the proof of Result 3.29) to prove the following distributive law for sets: Let A, B and C be sets. Then A (B C) = (A B) (A C).
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Chapter 3: Problem 15 Discrete Mathematics 1
Prove the following: Let A and B be sets. Then (A B) (B A) = (A B) (A B).
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Chapter 3: Problem 16 Discrete Mathematics 1
Let S = {0, 1, 3, 4}. Prove that if x S, then x2 4x + 3 = 0 or x2 4x + 3 = 3.
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