An Introduction to Mathematical Proofs

Logic

Propositions; Logical Connectives; Truth Tables

Logical Equivalence; IF-Statements

IF, IFF, Tautologies, and Contradictions

Tautologies; Quantifiers; Universes

Properties of Quantifiers: Useful Denials

Denial Practice; Uniqueness

Proofs

Definitions, Axioms, Theorems, and Proofs

Proving Existence Statements and IF Statements

Contrapositive Proofs; IFF Proofs

Proofs by Contradiction; OR Proofs

Proof by Cases; Disproofs

Proving Universal Statements; Multiple Quantifiers

More Quantifier Properties and Proofs (Optional)

Sets

Set Operations; Subset Proofs

More Subset Proofs; Set Equality Proofs

More Set Quality Proofs; Circle Proofs; Chain Proofs

Small Sets; Power Sets; Contrasting ? and ?

Ordered Pairs; Product Sets

General Unions and Intersections

Axiomatic Set Theory (Optional)

Integers

Recursive Definitions; Proofs by Induction

Induction Starting Anywhere: Backwards Induction

Strong Induction

Prime Numbers; Division with Remainder

Greatest Common Divisors; Euclid's GCD Algorithm

More on GCDs; Uniqueness of Prime Factorizations

Consequences of Prime Factorization (Optional)

Relations and Functions

Relations; Images of Sets under Relations

Inverses, Identity, and Composition of Relations

Properties of Relations

Definition of Functions

Examples of Functions; Proving Equality of Functions

Composition, Restriction, and Gluing

Direct Images and Preimages

Injective, Surjective, and Bijective Functions

Inverse Functions

Equivalence Relations and Partial Orders

Reflexive, Symmetric, and Transitive Relations

Equivalence Relations

Equivalence Classes

Set Partitions

Partially Ordered Sets

Equivalence Relations and Algebraic Structures (Optional)

Cardinality

Finite Sets

Countably Infinite Sets

Countable Sets

Uncountable Sets

Real Numbers (Optional)

Axioms for R; Properties of Addition

Algebraic Properties of Real Numbers

Natural Numbers, Integers, and Rational Numbers

Ordering, Absolute Value, and Distance

Greatest Elements, Least Upper Bounds, and Completeness

Suggestions for Further Reading