An introduction to mathematical proofs <P>Logic< P><P>Propositions; Logical Connectives; Truth Tables< P><P>Logical Equivalence; IF Statements< P><P>IF IFF Tautologies and Contradictions< P><P>Tautologies; Quantifiers; Universes< P><P>Properties of Quantifiers: Useful Denials< P><P>Denial Practice; Uniqueness< P><P>< P><B><P>Proofs< P>< B><P>Definitions Axioms Theorems and Proofs< P><P>Proving Existence Statements and IF Statements< P><P>Contrapositive Proofs; IFF Proofs< P><P>Proofs by Contradiction; OR Proofs< P><P>Proof by Cases; Disproofs< P><P>Proving Universal Statements; Multiple Quantifiers< P><P>More Quantifier Properties and Proofs (Optional)< P><P>< P><B><P>Sets< P>< B><P>Set Operations; Subset Proofs< P><P>More Subset Proofs; Set Equality Proofs< P><P>More Set Quality Proofs; Circle Proofs; Chain Proofs< P><P>Small Sets; Power Sets; Contrasting  and ⁶< P><P>Ordered Pairs; Product Sets< P><P>General Unions and Intersections< P><P>Axiomatic Set Theory (Optional)< P><P>< P><B><P>Integers< P>< B><P>Recursive Definitions; Proofs by Induction< P><P>Induction Starting Anywhere: Backwards Induction< P><P>Strong Induction< P><P>Prime Numbers; Division with Remainder< P><P>Greatest Common Divisors; Euclids GCD Algorithm< P><P>More on GCDs; Uniqueness of Prime Factorizations< P><P>Consequences of Prime Factorization (Optional)< P><P>< P><B><P>Relations and Functions< P>< B><P>Relations; Images of Sets under Relations< P><P>Inverses Identity and Composition of Relations< P><P>Properties of Relations< P><P>Definition of Functions< P><P>Examples of Functions; Proving Equality of Functions< P><P>Composition Restriction and Gluing< P><P>Direct Images and Preimages< P><P>Injective Surjective and Bijective Functions< P><P>Inverse Functions< P><P>< P><B><P>Equivalence Relations and Partial Orders< P>< B><P>Reflexive Symmetric and Transitive Relations< P><P>Equivalence Relations< P><P>Equivalence Classes< P><P>Set Partitions< P><P>Partially Ordered Sets< P><P>Equivalence Relations and Algebraic Structures (Optional)< P><P>< P><B><P>Cardinality< P>< B><P>Finite Sets< P><P>Countably Infinite Sets< P><P>Countable Sets< P><P>Uncountable Sets< P><P>< P><B><P>Real Numbers (Optional)< P>< B><P>Axioms for R; Properties of Addition< P><P>Algebraic Properties of Real Numbers< P><P>Natural Numbers Integers and Rational Numbers< P><P>Ordering Absolute Value and Distance< P><P>Greatest Elements Least Upper Bounds and Completeness< P><P>< P><B><P>Suggestions for Further Reading< P>< B>