MAT 310 BRIDGE TO ADVANCED MATHEMATICS

Richard HammackVirginia Commonwealth University

Virginia Commonwealth University

MAT 310 Bridge to Advanced Mathematics

Richard Hammack

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This text was compiled on 05/04/2022

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1 5/4/2022

TABLE OF CONTENTS

This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use toverify theorems, explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematicscourses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively aboutmathematics.

1: Sets

1.1: Introduction to Sets1.2: The Cartesian Product1.3: Subsets1.4: Power Sets1.5: Union, Intersection, Difference1.6: Complement1.7: Venn Diagrams1.8: Indexed Sets1.8: Sets That Are Number Systems1.9: Russell’s Paradox

2: Logic

2.0: Statement2.1: And, Or, Not2.2: Conditional Statements2.3: Biconditional Statements2.4: Truth Tables for Statements2.5: Logical Equivalence2.6: Quantifiers2.7: More on Conditional Statements2.8: Translating English to Symbolic Logic2.9: Negating Statements2.10: Logical Inference2.11: An Important Note

3: Counting

3.0: Lists3.1: The Multiplication Principle3.2: The Addition and Subtraction Principles3.3: Factorials and Permutations3.4: Counting Subsets3.5: Pascal’s Triangle and the Binomial Theorem3.6: The Inclusion-Exclusion Principle3.7: Counting Multisets3.8: New Page3.9: New Page

2 5/4/2022

4: Direct Proof

4.1: Section 1-4.2: Section 2-4.3: Section 3-4.4: Section 4-4.5: Section 5-4.6: Section 6-Exercises

5: Contrapositive Proof

5.1: Section 1-5.2: Section 2-5.3: Section 3-5.4: Section 4-5.5: Section 5-5.6: Section 6-

6: Proof by Contradiction

6.0: Section 1-6.1: Section 2-6.2: Section 3-6.3: Section 4-6.4: Section 5-6.5: Section 6-

7: Proving Non-Conditional Statements

7.0: Section 1-7.1: Section 2-7.2: Section 3-7.3: Section 4-7.4: Section 5-7.5: Section 6-

8: Proofs Involving Sets

8.0: Section 1-8.1: Section 2-8.2: Section 3-8.3: Section 4-

9: Disproof

9.0: Section 1-9.1: Section 2-9.2: Section 3-9.3: Section 4-9.4: Section 5-9.5: Section 6-

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10: Mathematical Induction

10.0: Section 1-10.1: Section 2-10.2: Section 3-10.3: Section 4-10.4: Section 5-10.5: Section 6-

11: Relations

11.0: Section 1-11.1: Section 2-11.2: Section 3-11.3: Section 4-11.4: Section 5-11.5: Section 6-

12: Functions

12.0: Section 1-12.1: Section 2-12.2: Section 3-12.3: Section 4-12.4: Section 5-12.5: Section 6-

13: Proofs in Calculus

13.0: Section 1-13.1: Section 2-13.2: Section 3-13.3: Section 4-13.4: Section 5-13.5: Section 6-

14: Cardinality of Sets

14.0: Section 1-14.1: Section 2-14.2: Section 3-14.3: Section 4-14.4: Section 5-14.5: Section 6-

Index

Glossary

Thumbnail: P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5. (Public Domain). Text from Oscar Levin's DiscreteMathematics text (CC BY-SA).

4 5/4/2022

MAT 310 Bridge to Advanced Mathematics is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by RichardHammack.

1 5/4/2022

CHAPTER OVERVIEW

1: Sets1.1: Introduction to Sets1.2: The Cartesian Product1.3: Subsets1.4: Power Sets1.5: Union, Intersection, Difference1.6: Complement1.7: Venn Diagrams1.8: Indexed Sets1.8: Sets That Are Number Systems1.9: Russell’s Paradox

1: Sets is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Richard Hammack.

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1.1: Introduction to Sets

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1.2: The Cartesian Product

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1.3: Subsets

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1.4: Power Sets

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1.5: Union, Intersection, Difference

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1.6: Complement

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1.7: Venn Diagrams

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1.8: Indexed Sets

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1.8: Sets That Are Number Systems

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1.9: Russell’s Paradox

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CHAPTER OVERVIEW

2: Logic2.0: Statement2.1: And, Or, Not2.2: Conditional Statements2.3: Biconditional Statements2.4: Truth Tables for Statements2.5: Logical Equivalence2.6: Quantifiers2.7: More on Conditional Statements2.8: Translating English to Symbolic Logic2.9: Negating Statements2.10: Logical Inference2.11: An Important Note

2: Logic is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Richard Hammack.

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2.0: Statement

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2.1: And, Or, Not

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2.2: Conditional Statements

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2.3: Biconditional Statements

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2.4: Truth Tables for Statements

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2.5: Logical Equivalence

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2.6: Quantifiers

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2.7: More on Conditional Statements

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2.8: Translating English to Symbolic Logic

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2.9: Negating Statements

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2.10: Logical Inference

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2.11: An Important Note

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CHAPTER OVERVIEW

3: Counting3.0: Lists3.1: The Multiplication Principle3.2: The Addition and Subtraction Principles3.3: Factorials and Permutations3.4: Counting Subsets3.5: Pascal’s Triangle and the Binomial Theorem3.6: The Inclusion-Exclusion Principle3.7: Counting Multisets3.8: New Page3.9: New Page

3: Counting is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Richard Hammack.

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3.0: Lists

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3.1: The Multiplication Principle

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3.2: The Addition and Subtraction Principles

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3.3: Factorials and Permutations

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3.4: Counting Subsets

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3.5: Pascal’s Triangle and the Binomial Theorem

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3.6: The Inclusion-Exclusion Principle

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3.7: Counting Multisets

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3.8: New Page

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3.9: New Page

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CHAPTER OVERVIEW

4: Direct Proof4.1: Section 1-4.2: Section 2-4.3: Section 3-4.4: Section 4-4.5: Section 5-4.6: Section 6-Exercises

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4.1: Section 1-

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4.2: Section 2-

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4.3: Section 3-

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4.4: Section 4-

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4.5: Section 5-

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4.6: Section 6-ExercisesUse the method of direct proof to prove the following statements.

Exercise If x is an even integer, then is even.

Exercise If x is an odd integer, then is odd.

Exercise If a is an odd integer, then is odd.

Exercise

Suppose . If x and y are odd, then xy is odd.

Exercise Suppose . If x is even, then xy is even.

Exercise Suppose . If and , then .

Exercise Suppose . If , then .

Exercise

Suppose a is an integer. If , then .

Exercise

Suppose a is an integer. If , then .

Exercise Suppose a and b are integers. If , then .

Exercise Suppose . If and , then .

Exercise

If and , then .

Exercise Suppose . If , then or .

Exercise If , then is odd. (Trycases.)

Exercise

If , then is even. (Trycases.)

4.6.1

x2

4.6.2

x3

4.6.3

+3a+5a2

4.6.4

x, y ∈ Z

4.6.5

x, y ∈ Z

4.6.6

a, b, c ∈ Z a|b a|c a|(b+c)

4.6.7

a, b ∈ Z a|b |a2 b2

4.6.8

5|2a 5|a

4.6.9

7|4a 7|a

4.6.10

a|b a|(3 −b2 +5b)b3

4.6.11

a, b, c, d ∈ Z a|b c|d ac|bd

4.6.12

x ∈ R 0 < x < 4 ≥ 14x(4−x)

4.6.13

x, y ∈ R +5y = +5xx2 y2 x = y x+y = 5

4.6.14

n ∈ Z 5 +3n+7n2

4.6.15

n ∈ Z +3n+4n2

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Exercise If two integers have the same parity, then their sum is even. (Try cases.)

Exercise If two integers have opposite parity, then their product is even.

Exercise Suppose x and y are positive real numbers. If , then .

Exercise

Suppose a, b and c are integers. If and , then .

Exercise

If a is an integer and , then .

Exercise If p is prime and k is an integer for which , then p divides .

Exercise

If , then . (You may need a separate case for .)

Exercise

If , then is even.

Exercise If and , then the numbers are all composite. (Thus for any , onecan find consecutive composite numbers. This means there are arbitrarily large “gaps” between prime numbers.)

Exercise

If and , then .

Exercise

Every odd integer is a difference of two squares. (Example , etc.)

Exercise Suppose . If , then or b is not prime.

Exercise

Let . Suppose a and b are not both zero, and . Prove that .

4.6: Section 6-Exercises is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Richard Hammack.

4.6.16

4.6.17

4.6.18

x < y <x2 y2

4.6.19

|ba2 |cb3 |ca6

4.6.20

|aa2 a ∈ {−1, 0, 1}

4.6.21

0 < k < p ( )pk

4.6.22

n ∈ N = 2( )+( )n2 n2

n1 n = 1

4.6.23

n ∈ N ( )2nn

4.6.24

n ∈ N n ≥ 2 n! +2,n! +3,n! +4,n! +5, ⋯ ,n! +n n ≥ 2

n−1

4.6.25

a, b, c ∈ N c ≤ b ≤ a ( )( ) = ( )( )a

b

a

c

a

b−c

a−b+c

c

4.6.26

7 = −42 32

4.6.27

a, b ∈ N gcd(a, b) > 1 b|a

4.6.28

a, b, c ∈ Z c ≠ 0 c ⋅ gcd(a, b) ≤ gcd(ca, cb)

1 5/4/2022

CHAPTER OVERVIEW

5: Contrapositive Proof5.1: Section 1-5.2: Section 2-5.3: Section 3-5.4: Section 4-5.5: Section 5-5.6: Section 6-

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5.1: Section 1-

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5.2: Section 2-

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5.3: Section 3-

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5.4: Section 4-A. Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In mostcases contrapositive is easier.)

Exercise Suppose . If is even, then n is even.

Exercise Suppose . If is odd, then n is odd.

Exercise Suppose . If is odd, then a and b are odd.

Exercise

Suppose . If a does not divide bc, then a does not divide b.

Exercise Suppose . If then .

Exercise Suppose . If then .

Exercise Suppose . If both ab and are even, then both a and b are even.

Exercise

Suppose . If , then .

Exercise

Suppose . If , then .

Exercise Suppose and . If , then and .

Exercise Suppose . If is even, then x is even or y is odd.

Exercise Suppose . If is not divisible by 4, then a is odd.

Exercise

Suppose . If , then .

B. Prove the following statements using either direct or contrapositive proof.

Exercise If and a and b have the same parity, then and do not.

5.4.1

n ∈ Z n2

5.4.2

n ∈ Z n2

5.4.3

a, b ∈ Z ( −2b)a2 b2

5.4.4

a, b, c ∈ Z

5.4.5

x ∈ R +5x < 0x2 x < 0

5.4.6

x ∈ R −x > 0x3 x > −1

5.4.7

a, b ∈ Z a +b

5.4.8

x ∈ R −4 +3 − +3x −4 ≥ 0x5 x4 x3 x2 x ≥ 0

5.4.9

n ∈ Z 3�� ∤ n2 3�� ∤ n

5.4.10

x, y, z ∈ Z x ≠ 0 x ∤ ��yz x ∤ ��y x ∤ ��z

5.4.11

x, y ∈ Z (y +3)x2

5.4.12

a ∈ Z a2

5.4.13

x ∈ R +7 +5x ≥ + +8x5 x3 x4 x2 x ≥ 0

5.4.14

a, b ∈ Z 3a +7 7b −4

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Exercise Suppose . If is even, then x is odd.

Exercise Suppose . If is even, then x and y have the same parity.

Exercise If n is odd, then .

Exercise

If , then .

Exercise

Let and . If and , then .

Exercise If and , then .

Exercise Let and . If , then .

14. Let a∈Z, n∈N. If a has remainder r when divided by n, then a≡r (mod n).

15. Leta,b∈Zandn∈N.Ifa≡b(modn),thena2≡ab(modn).

16. If a≡b (mod n) and c≡d (mod n), then ac≡bd (mod n).

17. Letn∈N.If2n−1isprime,thennisprime.

18. Ifn=2k−1fork∈N,theneveryentryinRownofPascal’sTriangleisodd.

19. If a≡0 (mod 4) or a≡1 (mod 4), then ��a2�� is even.

20. Ifn∈Z,then4��(n2−3).

21. Ifintegersaandbarenotbothzero,thengcd(a,b)=gcd(a−b,b).

22. Ifa≡b(modn),thengcd(a,n)=gcd(b,n).

23. Suppose the division algorithm applied to a and b yields a = qb + r. Prove gcd(a, b) = gcd(r, b).

24. If a ≡ b (mod n), then a and b have the same remainder when divided by n

5.4.15

x ∈ Z −1x3

5.4.16

x, y ∈ Z x +y

5.4.17

8|( −1)n2

5.4.18

a, b ∈ Z (a +b ≡ + (mod 3))3 a3 b3

5.4.19

a, b, c ∈ Z n ∈ N a ≡ b (mod n) a ≡ c (mod n) c ≡ b (mod n)

5.4.20

a ∈ Z a ≡ 1 (mod 5) ≡ 1 (mod 5)a2

5.4.21

a, b ∈ Z n ∈ N a ≡ b (mod n) ≡ (mod n)a3 b3

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5.5: Section 5-

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CHAPTER OVERVIEW

6: Proof by Contradiction6.0: Section 1-6.1: Section 2-6.2: Section 3-6.3: Section 4-6.4: Section 5-6.5: Section 6-

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7: Proving Non-Conditional Statements7.0: Section 1-7.1: Section 2-7.2: Section 3-7.3: Section 4-7.4: Section 5-7.5: Section 6-

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CHAPTER OVERVIEW

8: Proofs Involving Sets8.0: Section 1-8.1: Section 2-8.2: Section 3-8.3: Section 4-

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CHAPTER OVERVIEW

9: Disproof9.0: Section 1-9.1: Section 2-9.2: Section 3-9.3: Section 4-9.4: Section 5-9.5: Section 6-

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9.5: Section 6-

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10: Mathematical Induction10.0: Section 1-10.1: Section 2-10.2: Section 3-10.3: Section 4-10.4: Section 5-10.5: Section 6-

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10.5: Section 6-

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11: Relations11.0: Section 1-11.1: Section 2-11.2: Section 3-11.3: Section 4-11.4: Section 5-11.5: Section 6-

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12: Functions12.0: Section 1-12.1: Section 2-12.2: Section 3-12.3: Section 4-12.4: Section 5-12.5: Section 6-

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CHAPTER OVERVIEW

13: Proofs in Calculus13.0: Section 1-13.1: Section 2-13.2: Section 3-13.3: Section 4-13.4: Section 5-13.5: Section 6-

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CHAPTER OVERVIEW

14: Cardinality of Sets14.0: Section 1-14.1: Section 2-14.2: Section 3-14.3: Section 4-14.4: Section 5-14.5: Section 6-

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IndexAaddition principle

3.2: The Addition and Subtraction Principles

BBiconditional Statement

2.3: Biconditional Statements binomial theorem

3.5: Pascal’s Triangle and the Binomial Theorem

Ddifference

1.5: Union, Intersection, Difference

Eelements

1.1: Introduction to Sets

Ffactorial

3.3: Factorials and Permutations

Iindexed sets

1.8: Indexed Sets intersection

1.5: Union, Intersection, Difference

Oordered pairs

1.2: The Cartesian Product

PPascal's Triangle

3.5: Pascal’s Triangle and the Binomial Theorem permutations

3.3: Factorials and Permutations Power Sets

1.4: Power Sets

Proof by Contradiction6: Proof by Contradiction

Sset

1.1: Introduction to Sets subtraction principle

3.2: The Addition and Subtraction Principles

TTruth Table

2.4: Truth Tables for Statements

Uunion

1.5: Union, Intersection, Difference

VVenn diagrams

1.7: Venn Diagrams

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GlossarySample Word 1 | Sample Definition 1