# Math 109: Outline of the Course

## 1 Preliminaries

1.1 Set Theory

Definitions Set, element, empty set, cardinality/order, subset/inclusion, containment, power set , intersection,
union, complement

Theorems DeMorgan’ s Laws
Know How to Prove Set inclusion, Set Equality
Compute Set operations, cardinality of finite sets, power set

1.2 Functions

Definitions Function, domain, image of a set under a function, injective/one-to-one, surjective/onto, bijective,
composition, inverse, permutation
Know How to Prove Injective, surjective, bijective
Compute Image of a set, composition of functions, composition of permutations, inverses of permutations

1.3 Equivalence Relations

Definitions Equivalence relation, equivalence class, quotient of a set by ~
Know How to Prove Equivalence relation
Compute Equivalence classes, quotients

## 2 Proofs

Definitions Propositional statement (if p then q), if and only if, converse, negation of a statement, contrapositive
Theorems A propositional statement is true if and only if its contra positive is true
Know How to Prove If/then statements, if and only if statements, proof by contradiction, proof by contrapositive,
induction
Compute Negation of a statement

## 3 Number Theory

Definitions Divides, factor , prime, even, odd, gcd, lcm , rational number

Theorems
Division Theorem
* There exist infinitely many primes
* is irrational
Fundamental Theorem of Arithmetic
Compute gcd(a, b) and m and n such that am + bn = gcd(a, b) (Euclidean Algorithm)

## 4 Metric Spaces

Definitions Metric space, metric, triangle inequality , sup, inf, isometry, open set, neighborhood, closed set,
limit, closure, dense, continuous, Cauchy sequence, complete

Theorems

An open (alt. closed) ball is open (closed)
An arbitrary union of open sets is open
A finite intersection of open sets is open
* An arbitrary intersection of closed sets is closed
* A finite union of closed sets is closed
* If
* A sequence can have only one limit (hw)
f : X -> Y is continuous at x0 if and only if for every
f : X -> Y is continuous if and only if for every open (alt. closed) set V in Y , is open (closed) in X
* A convergent sequence is Cauchy
R is complete
* A closed subset of a complete metric space is complete

Know How to Prove A set is a metric space, a set is open, a set is closed, a sequence converges to a limit,
a function is continuous, a sequence is Cauchy

Examples
Metric Spaces: Standard metric on Rn, discrete metric
Continuous Functions: polynomials
Complete Metric Spaces: R, any closed set in R

## 5 Topology

Definitions Topo logical space , topology, open, closed, neighborhood, continuous, homeomorphism
Know How to Prove A collection of sets is a topology
Examples Any metric space, some other strange ones

## 6 Group Theory

Definitions Binary operation, closed with respect to *, associative, identity, inverse, commutative, group,
abelian, subgroup, homomorphism, isomorphism

Theorems
* An identity element with respect to * is unique
* If * is associative, inverses are unique
* Then inverse of (a*b) is b−1a−1

Know How to Prove An operation is associative , an element is the identity, an element is an inverse to
another element, an operation is commutative, a map is a homomorphism or isomorphism

Compute Multiplication tables , subgroups, order of elements

Examples

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