Discrete Mathematics Notes
Kevin Shu
September 1, 2023
August 30, 2023
1
What on Earth even is set theory?
Simple intro to set theory ideas
In essence, think of what mathematics is for. At its base level, it is used to represent objects in the real world - to know that
they exist, to group them by a shared property, to quantify them, and to understand what operations may be applied to them (i.e,
addition, subtraction, etc.). This is the base of the set theory idea of mathematics, which seeks to formalize this idea. However,
various paradoxes can arise when these very loose definitions are used. These kinds of definitions of set theory are called naive set
theory. To combat this, the Zermelo–Fraenkel set theory uses rules and notations are used to lay out several axioms.
2
Truth and Provability
Difference between truth and provability
Imagine we have a rock. Could be radioactive, you use Geiger counter to measure. The Geiger counter doesn’t read anything. Is
the rock radioactive?
The Geiger counter is the proof. The rock could in fact be radioactive, but as far as we know it is not. Nature knows truth, we
know provability. True statements are not always provable, provable statements are not always true.
”Mathematically Sound” - means that I can follow a chain of simplifying concepts to an end. Think of a concept. Now think of
a way to explain that concept with simpler terms. Now think of a way to explain those concepts with simpler terms. When there
is an end to this chain, the concept is mathematically sound.
3
The Axioms of Set Theory
Equality
1. (Extensionality) Two sets are equal if and only if they have the same elements.
x = {A, B} = {B, A} = {A, A, B}, {} =
̸ {{}}
(∀x)(∀y)(x = y) ↔ (∀z)((z ∈ x) ↔ (z ∈ y))
Existence of Special Sets
2. (Empty Set) There is a set with no elements.
{} = ∅ = 0
Reworded: There exists x such that for any y, y is not in x
(∃x)(∀y)¬(y ∈ x)
3. (Infinity) There is an inductive set.
Creation of new Sets - Ways to create new sets
4. (Pairing) Given sets A and B (or A and A!), there is a set whose only elements are A and B. (I.E., the unordered pair A, B
is a set.)
P = {A, B} or Q = {A}
Reworded: For all A and B, there exists a set P where, for all z in P, z is either A or B
(∀A)(∀B)(∃P )(∀z)((z ∈ P ) ↔ (z = A) ∨ (z = B))
5. (Union) If A is a set, then there is a set ∪A, called the union of A, whose elements are the elements of the elements of A
A = {x, y, z}, B = {y, x, w}, C = {u, z}
Infix notation: A ∪ B = {x, y, z} ∪ {y, x, w}
Prefix notation: ∪{A, B} = A ∪ B
For all A (this is a set of sets), there exists U (this is the united set), such that for all z, z exists in U if and only if there
exists w such that z is in w and w is in A
(∀A)(∃U )(∀z)((z ∈ U ) ↔ (∃w)((z ∈ u)(w ∈ A)))
4
Symbols
The language of set theory
ˆ ∧ - ”and”
ˆ ∨ - ”or”
ˆ → - ”implies”
ˆ ↔ - ”if and only if”, ”equivalent”
ˆ ∀ - ”for all”
ˆ ∃ - ”there exists”
ˆ ε - ”epsilon”
ˆ δ - ”delta”
ˆ ∈ - ”in”, ”element of” (this is the most basic concept in set theory)
Defined symbols
ˆ ∅ - ”Empty Set” - Φ∅ (x) : ”(∀y)(¬(y ∈ x))”
ˆ ⊆ - ”Subset” - Φ⊆ (x, y) : ”(∀z)((z ∈ x) → (z ∈ y))”
ˆ P (x) - ”Power Set” - Φ( y = P (x))(x, y) : (∀z)((z ∈ y) ↔ (z ⊆ x))”
(Expanding on this) A = x, y, z, subsets of A:
∅, x, y, z, x, y, x, z, y, z, x, y, z