MTH307: Introduction to Mathematical Rigor
Fall 2016

## Syllabus

Here is a basic outline of the topics we will cover (time permitting!) this semester.
1. Fundamental Concepts
• Sets: Describing sets, products, unions, intersections and complements, Venn diagrams.
• Logic: Statements, compounds, conditional statements, quantifiers, negations.
2. Proving conditional statements
• Direct proof: Theorems and definitions, cases.
• Contrapositives: Proofs and exposition.
• Proof by contradiction: Proof and combining techniques in a proof.
3. More Proofs
• Non-conditional statements: Equivalences, existence and uniqueness.
• Proofs and sets: Set inclusion, subsets, equality.
• Induction: Mathematical induction, examples.
4. Relations, functions and cardinality
• Relations: Relations, equivalence relations, partitions.
• Functions: Injectivity, surjectivity, compositions, inverses, images.
• Cardinality: Equal cardinality, countability, Cantor-Schroder-Bernstein Theorem
• .

### Course Goals

By the end of the course, you should
• have a firm understanding of the fundamental conecepts in mathematics.
• Sets
• Relations
• Functions
• Cardinality
• understand the nature of mathematical proof, including the notion of statements, open sentences and conditional statements.
• be able to prove (simple) mathematical statements, particularly conditional statements, using techniques such as
• Direct Proof
• Contrapositive Proof