Mathematical Reasoning: Writing and Proof

Introduction to Writing Proofs

01What Is a Statement? Truth Values and If-Then 02Quantifiers and Counterexamples: What One Example Can Prove 03The Truth Table for If-Then: When a Conditional Is False 04Using Conditionals: Theorems, Counterexamples and Closure Problem set0/10 Problem set 20/10

01Even and Odd Integers: Why a Proof Starts From the Definition 02The Know-Show Table: Proving Odd Times Odd Is Odd 03Writing Proofs for a Reader: Six Guidelines and Sums of Even and Odd 04Backward Answers Must Imply the Goal: Two More Direct Proofs Problem set0/10 Problem set 20/10

Logical Reasoning

01Logical Operators: Truth Rules for And, Or, Not and If-Then 02Only If, Necessary and Sufficient: Truth Tables and Tautologies 03Logical Equivalence: De Morgan's Laws and the Contrapositive 04Negating If-Then and Proving Equivalences by Chains Problem set0/10 Problem set 20/10

01Sets and Variables: The Roster Method and the Universal Set 02Element, Subset and Truth Set: Open Sentences as Predicates 03Set Builder Notation, the Empty Set, and Sentences True for All x Problem set0/10 Problem set 20/10

01Quantifiers: When For Every and There Exists Statements Are True 02Negating Quantified Statements: The Quantifier Switches 03Counterexamples and Negating Definitions With Quantifiers 04Statements With Two Quantifiers: Why the Order Matters Problem set0/10 Problem set 20/10

Constructing Proofs

01Divisibility: Definition, Notation, and Non-Divisors 02Proving Divisibility Is Transitive: Axioms, Definitions, Proof 03A Divisor of Two Numbers Divides Their Sum, and Counterexamples 04Congruence Modulo n: A Proof Mod 8 and 7 Rules for Writing Proofs Problem set0/10 Problem set 20/10

01Proving the Contrapositive When a Direct Proof Stalls 02Writing a Proof by Contrapositive: ab = 0 and De Morgan 03Proving Biconditionals: Two Conditionals, Each Marked 04Constructive and Nonconstructive Proofs of Existence Problem set0/10 Problem set 20/10

Contradiction, Cases and Proof Writing

01Why Proof by Contradiction Works and How to Negate the Claim 02Writing a Proof by Contradiction: x/y + y/x and Negations 03Proving Nonexistence by Contradiction: Even Equals Odd 04Proof That √2 Is Irrational: Rational and Irrational Numbers Problem set0/10 Problem set 20/10

01Proof by Cases: Why It Works, and Why n² + n Is Even 02Choosing Exhaustive Cases and Writing a Case Proof 03Absolute Value by Cases: Proving |x| = A and |-x| = |x|04The Triangle Inequality by Cases: |x + y| ≤ |x| + |y|Problem set0/10 Problem set 20/10

01Many Ways to Write 27 = 4q + r, and a Mod 6 Proof 02The Division Algorithm: One Quotient, One Remainder 03Proof by Remainders: Why 3 Divides n³ - n 04Adding, Multiplying and Powering Congruences 05Congruence Is Reflexive, Symmetric and Transitive 06Each Integer Is Congruent to Its Remainder: Squares Mod 5 07Ruling Out Cases to Prove a² ≡ 1 (mod 6)Problem set0/10 Problem set 20/10

01Direct, Contrapositive or Contradiction: Choosing a Proof 02Proof by Cases, Equivalences, and Existence Proofs 03Writing a Proof for a Reader: Openings and Signposts 04Writing the Math in a Proof: Displays and a Clear Ending Problem set0/10 Problem set 20/10

Mathematical Induction

01Inductive Sets: Why a Table of Cases Is Not a Proof 02The Principle of Mathematical Induction as an Axiom 03Induction on Sums: Add the Next Term to Both Sides 04Why Both Steps Matter: Proving 4 Divides 5ⁿ − 1 Problem set0/10 Problem set 20/10

01When Ordinary Induction Falls Short: n! vs 2ⁿ and Primes 02Extended Induction: Start the Basis Step at Any Integer M 03Strong Induction: Every Integer Above 1 Factors into Primes 04Three Basis Cases: Every n from 8 On Is a Sum of 3s and 5s Problem set0/10 Problem set 20/10

01Defining Sequences by Recursion: From Halving to n!02Fibonacci Numbers: Patterns, Rabbits and Sanskrit Verse 03Every Third Fibonacci Number Is Even: An Induction Proof 04Geometric Series by Induction: a(1 − rⁿ)/(1 − r)Problem set0/10 Problem set 20/10

Set Theory

01Intersection, Union, Difference, Complement: And, Or, Not for Sets 02Venn Diagrams for Two Sets: Four Regions and De Morgan's Law 03Proper Subsets, the Empty Set and Set Equality 04Venn Diagrams for Three Sets: Eight Regions and Subset Conditions 05Power Sets Have 2ⁿ Elements: Cardinality and the Number Systems Problem set0/10 Problem set 20/10

01Choose-an-Element Proofs: Multiples of 6 Are a Subset of the Evens 02Complements Reverse Subsets: A ⊆ B Implies Bᶜ ⊆ Aᶜ 03Proving Set Equality by Two Inclusions: A − (A − B) = A ∩ B 04Proving a Set Is Empty: Disjoint Sets and Element Chasing Problem set0/10 Problem set 20/10

01De Morgan's Law for Unions and Cycles of Conditionals 02Laws of Union and Intersection: Proving A ∩ B = B ∩ A 03Distributive Law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)04Complement Laws and Proofs by the Algebra of Sets Problem set0/10 Problem set 20/10

01Ordered Pairs as Solutions: The Truth Set of 2x + 3y = 12 02The Cartesian Product A × B: Listing Pairs, Testing Laws 03Products of Intervals Are Rectangles in the Plane 04Proving A × (B ∪ C) = (A × B) ∪ (A × C)Problem set0/10 Problem set 20/10

01Union and Intersection of a Family: Some Versus All 02Indexed Unions: Subscripts for Infinitely Many Sets 03Families Indexed by Any Set: One Interval per Real Number 04De Morgan's Laws for Any Family, and Pairwise Disjoint Problem set0/10 Problem set 20/10

Functions

01What Is a Function? Exactly One Output for Each Input 02Functions Without Formulas: Birthdays and Sums of Divisors 03Images and Preimages: The Formal Definition of a Function 04The Range of a Function: From Graphs and Arrow Diagrams Problem set0/10 Problem set 20/10

01Polygon Diagonals and Derivatives: A Formula Is Not a Function 02Equal Functions: Why x⁵ mod 5 Is the Identity on R₅ 03Derivatives, Averages and Sequences as Functions 04Functions of Two Variables: Preimages in ℤ × ℤ Problem set0/10 Problem set 20/10

01Shared and Missed Outputs: Why 5x + 3 Has Neither 02Injections: Proving a Function Is One-to-One 03Surjections and Bijections: Is 3m + 2 Onto?04Domain and Codomain Decide: x² + 1 and e⁻ˣ 05One-to-One and Onto for Functions on ℝ × ℝ Problem set0/10 Problem set 20/10

01Following Two Arrows: How g(f(x)) Builds a Function 02g ∘ f: The Function on the Right Acts First 03Decomposing a Function: Cut at the Last Step 04Composing Surjections: Pull c Back Through g, Then f 05What an Injective or Surjective Composite Reveals Problem set0/10 Problem set 20/10

01A Function as a Set of Ordered Pairs: Each Input Once 02Reversing Every Arrow: The Function That Undoes a Bijection 03Swapping the Pairs: The Inverse of a Function as a Set 04Proof: f Inverse Is a Function Exactly When f Is a Bijection 05Undo the Last Step First: The Inverse of a Composite Problem set0/10 Problem set 20/10

01Applying a Function to a Set: Forward and Backward 02Image and Preimage: Why Squaring Splits an Interval 03Images, Preimages and Set Operations: Six Conjectures 04Proof: The Image of an Intersection Is Only a Subset 05Preimages Preserve Unions and Intersections Exactly Problem set0/10 Problem set 20/10

Relations

01Conditions on Pairs: Bordering States and an Ellipse 02Relations Defined: Any Subset of A × B, Domain and Range 03x R y Notation: Testing the Divides Relation Three Ways 04Which Relations Are Functions? The Circle vs the Parabola 05Drawing Relations: Shaded Regions and Directed Graphs Problem set0/10 Problem set 20/10

01Reflexive, Symmetric and Transitive Relations 02Congruence Modulo n as a Symmetric Relation 03Reading Relation Properties off a Directed Graph 04Equivalence Relations: All Three Properties 05Congruence Modulo n Means Matching Remainders 06Equivalence Relations: Angles, Lines and Set Sizes Problem set0/10 Problem set 20/10

Equivalence Relations: All Three Properties

Why is 'is a multiple of' not an equivalence relation, while a − b ∈ ℤ on the rationals is?

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Your summary note

1. Definition of an equivalence relation
  State the definition for a relation $\sim$ on a nonempty set $A$ and the phrase " $x$ is equivalent to $y$ ".
2. Logical form of each property
  Record that reflexivity is a universal statement proved for an arbitrary element, while symmetry and transitivity are conditionals proved by assuming the hypothesis.
3. The relation "is a multiple of" on $\mathbb{Z}$
  Prove reflexivity and transitivity with $x = y \cdot p$ , $y = z \cdot q$ , and give the witness $4$ , $2$ against symmetry.
4. $x \sim y \iff x - y \in \mathbb{Z}$ on $\mathbb{Q}$
  Test the pairs $\frac{3}{4}, \frac{7}{4}$ and $\frac{3}{4}, \frac{1}{2}$ , then write the three proofs using $x - x = 0$ , negation and $(x - y) + (y - z)$ .
Attempt 1 of 2