An Introduction to Mathematical Proofs (Hardcover, 1)

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