1. The Logic of Mathematical Proofs:
a. Propositions and connectives
b. Conditionals and biconditionals
c. Quantifiers
d. Mathematical proofs
e. Proofs involving quantifiers
f. Induction
2. Set Theory
a. Notation and set theory
b. Set operations
c. DeMorgan's Laws
3. Functions
a. Cartesian Products
b. Relations
c. Equivalence relations
d. Partitions
e. Functions
f. Onto functions
g. One-to-one functions
h. Induced set functions
3. Cardinality
a. Equivalent sets
b. Finite sets
c. Countable sets
d. Uncountable sets
e. Schroder-Bernstein Theorem, the Axiom of Choice, and Zorn's Lemma
4. The Real Numbers
a. Properties of real numbers
b. completeness of real numbers
c. The Heine-Borel Theorem
d. The Bolzano-Weierstrass Theorem
1. Prove that any bounded subset of real numbers without an accumulation point is finite.
2. Let R be an equivalence relation. Show that xRy if xR=yR