Math 2001: Introduction to Discrete Mathematics

126 Pages·2019·0.408 MB·English

by  Agnès Beaudry

Math 2001 – Introduction to Discrete
Mathematics
Agn`es Beaudry
June 24, 2019
Contents
1 Sets 6
1.1 Monday, January 17 : First Day of Class . . . . . . . . . . . . 6
1.1.1 Sets and Subsets . . . . . . . . . . . . . . . . . . . . . 7
1.2 Friday, January 19 . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . 10
1.3 Monday, January 22 . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Set-builder notation . . . . . . . . . . . . . . . . . . . 12
1.3.2 Cardinality, informally . . . . . . . . . . . . . . . . . . 12
1.3.3 The Cartesian Product . . . . . . . . . . . . . . . . . . 13
1.4 Wednesday, January 24 . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Cardinality Continued . . . . . . . . . . . . . . . . . . 16
1.4.2 Note on the empty set . . . . . . . . . . . . . . . . . . 16
1.4.3 Power Set . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Friday, January 26 . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.1 Intersections, Unions, Difference and Complements . . 18
1.5.2 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Monday, January 29 . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.1 Index Sets . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Logic 25
2.1 Wednesday, January 31 . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 And, Or . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.3 Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Friday, February 2 . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 LaTeX . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Monday, February 5 . . . . . . . . . . . . . . . . . . . . . . . 31
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2.3.1 Conditional Statements . . . . . . . . . . . . . . . . . . 31
2.4 Wednesday, February 7 . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 The Converse . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.2 The Contrapositive . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Biconditional Statements . . . . . . . . . . . . . . . . . 37
2.4.4 Negating statements . . . . . . . . . . . . . . . . . . . 38
2.5 Friday, February 9 . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Monday, February 12 . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.2 Quantifyers . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.3 Logical Inference . . . . . . . . . . . . . . . . . . . . . 44
2.7 Wednesday, February 14 . . . . . . . . . . . . . . . . . . . . . 46
2.7.1 Broad Strokes for the Topic of Proofs . . . . . . . . . . 46
2.8 Friday, February 16 . . . . . . . . . . . . . . . . . . . . . . . . 48
2.9 Monday, February 19 . . . . . . . . . . . . . . . . . . . . . . . 49
2.9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 50
2.9.2 Examples of Direct and Contrapositive Proofs . . . . . 51
2.10 Wednesday, February 21 . . . . . . . . . . . . . . . . . . . . . 52
2.10.1 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.10.2 Without Loss of Generality . . . . . . . . . . . . . . . 54
2.11 Friday, February 23 . . . . . . . . . . . . . . . . . . . . . . . . 56
2.11.1 Without Loss of generality continued . . . . . . . . . . 56
2.11.2 Proof by Contradiction . . . . . . . . . . . . . . . . . . 57
2.12 Monday, February 26 . . . . . . . . . . . . . . . . . . . . . . . 59
2.12.1 Proof by Contradiction Continued . . . . . . . . . . . . 59
2.12.2 Congruence of Integers . . . . . . . . . . . . . . . . . . 60
2.13 Wednesday, February 28 . . . . . . . . . . . . . . . . . . . . . 61
2.13.1 If and only if, and the following are equivalent. . . . . . 61
2.14 Friday, March 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.15 Monday, March 5 . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.16 Wednesday, March 7 . . . . . . . . . . . . . . . . . . . . . . . 65
2.16.1 The following are equivalent (TFAE) . . . . . . . . . . 65
2.16.2 Existence Proofs . . . . . . . . . . . . . . . . . . . . . 67
2.16.3 Existence and Uniqueness Proofs . . . . . . . . . . . . 68
2.17 Friday, March 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.17.1 Proofs with sets . . . . . . . . . . . . . . . . . . . . . . 70
2.17.2 Proving x ∈ X . . . . . . . . . . . . . . . . . . . . . . 70
2.17.3 Proving X ⊆ Y . . . . . . . . . . . . . . . . . . . . . . 71
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2.17.4 Proving X (cid:54)⊆ Y . . . . . . . . . . . . . . . . . . . . . . 72
2.17.5 Proving X = Y . . . . . . . . . . . . . . . . . . . . . . 73
2.18 Monday, March 12 . . . . . . . . . . . . . . . . . . . . . . . . 76
2.18.1 (Weak) Mathematical Induction . . . . . . . . . . . . . 76
2.19 Wednesday, March 14 . . . . . . . . . . . . . . . . . . . . . . . 81
2.19.1 Weak Induction Continued . . . . . . . . . . . . . . . . 81
2.19.2 Strong Mathematical Induction Continued . . . . . . . 81
2.20 Friday, March 16 . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.21 Monday, March 19 . . . . . . . . . . . . . . . . . . . . . . . . 85
2.22 Wednesday, March 21 . . . . . . . . . . . . . . . . . . . . . . . 85
2.23 Friday, March 23 . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.24 Monday, April 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.24.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.24.2 Equivalence Relations . . . . . . . . . . . . . . . . . . 87
2.25 Wednesday, April 4 . . . . . . . . . . . . . . . . . . . . . . . . 89
2.25.1 Equivalence classes . . . . . . . . . . . . . . . . . . . . 89
2.26 Friday, April 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.27 Monday, April 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.27.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.28 Wednesday, April 11 . . . . . . . . . . . . . . . . . . . . . . . 95
2.28.1 Partitions, continued . . . . . . . . . . . . . . . . . . . 95
2.28.2 Integers modulo n . . . . . . . . . . . . . . . . . . . . . 96
2.29 Friday, April 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.29.1 Integers modulo n finished . . . . . . . . . . . . . . . . 98
2.30 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.31 Monday, April 16 . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.31.1 Functions continued . . . . . . . . . . . . . . . . . . . 101
2.31.2 Injectivity and Surjectivity . . . . . . . . . . . . . . . . 102
2.32 Wednesday, April 18 . . . . . . . . . . . . . . . . . . . . . . . 104
2.32.1 Injectivity and Surjectivity . . . . . . . . . . . . . . . . 104
2.33 Friday, April 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.33.1 The Pigeon-Hole Principle . . . . . . . . . . . . . . . . 107
2.34 Monday, April 23 . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.34.1 Image and Preimage . . . . . . . . . . . . . . . . . . . 109
2.34.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . 110
2.35 Wednesday, April 25 . . . . . . . . . . . . . . . . . . . . . . . 112
2.35.1 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.35.2 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . 113
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2.36 Friday, April 27 . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.36.1 Cardinality Continued . . . . . . . . . . . . . . . . . . 115
2.37 Monday, April 30 . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.37.1 Cardinality Continued . . . . . . . . . . . . . . . . . . 117
2.38 Wednesday, May 2 . . . . . . . . . . . . . . . . . . . . . . . . 122
2.38.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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Disclaimer. The following notes are my course notes, and may
contains mistakes and typos. They are shamelessly based on
book for the course:
• Book of Proof by Richard Hammack
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Chapter 1
Sets
1.1 Monday, January 17 : First Day of Class
Intro to discrete mathematics is a class designed to teach you the
language of higher mathematics. Sometimes, it may be more like
a language class than a math class. We will learn the rules of
logic, how to make mathematical statements and how to prove
them.
We will answer questions like: what does it mean to do mathe-
matics? What does it mean to prove something? You will learn
about some open problems and various areas.
Textbook: The primary source will distributed electronically on the
website as the course progresses. The official textbook is
• Book of Proof, by Richard Hammack
which available for free online.
Topics: Below is a list of the topics we will cover :
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• Basic Set Theory
• Basic Logic
• Mathematical Proofs
• Mathematical Induction
• Counting Techniques
• Relations and Functions
• Cardinality
1.1.1 Sets and Subsets
Definition 1.1 (Cantor’s Na¨ıve definition, 1882). “A set is a gathering to-
gether into a whole of definite, distinct objects of our perception or of our
thought — which are called elements of the set.”
So, a set is a “bunch of things”.
Notation. If A is a set and x is an element of A, we write
x ∈ A or A (cid:51) x.
If x is not an element of the set A, we write
x (cid:54)∈ A or A (cid:54)(cid:51) x.
The symbols { and } are used to denote a set. For example {a,b,c} is a
set containing three elements called a, b and c.
Example 1.2. • {0,1}
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• N = {1,2,3,...}
• Z = {0,1,−1,2,−2,...}
• X = {A,B,C,..., X,Y,Z}
• ∅ = { }. This is the set with no elements, called the empty set.
Definition 1.3. Suppose that A and B are sets. We say that A = B if the
elements of A are the same as the elements of B.
Warning 1.4. Sets don’t have repetitions and are not ordered.
• {a,b,c} = {c,b,a}
• {1,1,2,3} = {1,2,3} = {2,3,1}
Question 1.5. Is {0,1} equal to {0,{1}}?
Answer. No, these sets do not contain the same elements. Although they
both contain 0, the first set contains 1 while the second set contains the “set
containing 1”. Here is an imperfect analogy: 1 is to {1} as a dog is to a
picture of a dog.
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1.2 Friday, January 19
1.2.1 Subsets
Definition 1.6. A set A is a subset of B if all of the elements of A are also
elements of B.
Notation. If A is a subset of B, we write A ⊆ B. If A ⊆ B but A (cid:54)= B,
then sometimes write A (cid:40) B. If A is not a subset of B, we write A (cid:54)⊆ B.
Example 1.7. • {a,b} ⊆ {a,b,c}
• {a,b} (cid:40) {a,b,c}
• {a,b,c} ⊆ {a,b,c}
• N ⊆ Z
• N (cid:40) Z
• {z} (cid:54)⊆ {a,b,c}
We ended last class with the question of whether or not the set {0,1} is
equal to {0,{1}}. And this brought up another good question.
Question 1.8. Are elements of a set always also subsets of that set?
We came to the conclusion that the answer to both questions was NO.
Intuitively, a rock and a box containing a rock are different. One is a rock,
the other is something that contains a rock, but they are not the same thing.
Remark 1.9. Saying that the answer to Question 1.8 is NO does not mean
that there cannot be examples when this happens:
Question 1.10. Are there elements of the set {1,{1}} which are also subsets
of that set?
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