Proofs 101

1. Logic. 1.1 Introduction. 1.2. Statements and Logical Connectives. 1.3 Logical Equivalence. 1.4. Predicates and Quantifiers. 1.5. Negation. 2. Proof Techniques. 2.1. Introduction. 2.2. The Axiomatic and Rigorous Nature of Mathematics. 2.3. Foundations. 2.4. Direct Proof. 2.5. Proof by Contrapositive. 2.5. Proof by Cases. 2.6. Proof by Contradiction. 3. Sets. 3.1. The Concept of a Set. 3.2. Subset of Set Equality. 3.3. Operations on Sets. 3.4. Indexed Sets. 3.5. Russel’s Paradox. 4. Proof by Mathematical Induction. 4.1. Introduction. 4.2. The Principle of Mathematical Induction. 4.3. Proof by strong Induction. 5. Relations. 5.1. Introduction. 5.2. Properties of Relations. 5.3. Equivalence Relations. 6. Introduction. 6.1. Definition of a Function. 6.2. One-To-One and Onto Functions. 6.3. Composition of Functions. 6.4. Inverse of a Function. 7. Cardinality of Sets. 7.1. Introduction. 7.2. Sets with the same Cardinality. 7.3. Finite and Infinite Sets. 7.4. Countably Infinite Sets. 7.5. Uncountable Sets. 7.6 Comparing Cardinalities.