Here is a basic outline of the topics we will cover (time permitting!) this semester.
- Fundamental Concepts
- Sets: Describing sets, products, unions, intersections and complements, Venn diagrams.
- Logic: Statements, compounds, conditional statements, quantifiers, negations.
- Proving conditional statements
- Direct proof: Theorems and definitions, cases.
- Contrapositives: Proofs and exposition.
- Proof by contradiction: Proof and combining techniques in a proof.
- More Proofs
- Non-conditional statements: Equivalences, existence and uniqueness.
- Proofs and sets: Set inclusion, subsets, equality.
- Disproof: Counterexamples and contradiction.
- Induction: Mathematical induction, examples.
- Relations, functions and cardinality
- Relations: Relations, equivalence relations, partitions.
- Functions: Injectivity, surjectivity, compositions, inverses, images.
- Cardinality: Equal cardinality, countability, Cantor-Schroder-Bernstein Theorem.
By the end of the course, you should
- have a firm understanding of the fundamental conecepts in mathematics.
- 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
as well as being able to prove existence and uniqueness statements. Also to be able to use mathematical induction.
- Direct Proof
- Contrapositive Proof
- Proof by Contradiction
- perhaps most importantly of all, come to find mathematics as a subject of great beauty, far from the computational drudgery that is found in earlier courses. You will hopefully come to appreciate that the mathematical world is open ended, full of puzzles and questions, and some of you may even feel inspired to consider further mathematics courses which build on the techniques of this course.