Mathematics - Part 2: Set Theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.

collections of objects

Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with sets relevant to mathematics.

A set is a collection of objects (elements).

These elements can be numbers, symbols, other sets, etc.

1. Definition and Notation

What is a Set?

A set is a well-defined collection of distinct elements.

Example 1. Two sets defined:

A = {1, 2, 3}

B = {a, b, c}

Membership Symbols

• ∈ means "is an element of": 2 ∈ A

• ∉ means "is not an element of": 4 ∉ A

Set Builder Notation

Used when listing all elements is impractical.

P = {p | p is prime}

2. Equality of Sets

Two sets are equal if they contain the same elements, regardless of order or repetition.

A = {1, 2, 3}, B = {3, 1, 2, 2} ⇒ A = B

3. Cardinality (Size of a Set)

The cardinality of a set is the number of elements in it.

Denoted as |A|

|A| = 3

Example 2. Infinite set example

The set of prime numbers has cardinality ∞

4. Subsets

Set A is a subset of B if every element of A is also in B.

A ⊆ B

Proper Subset

If A is a subset of B but not equal to B:

A ⊂ B

5. The Empty Set

The empty set is a set with no elements.

∅ or {}

Properties

• ∅ ⊆ A for any set A

• Unique: there is only one empty set

6. Union and Intersection

Union

All elements from both sets:

A ∪ B = {x | x ∈ A or x ∈ B}

Intersection

Only elements common to both sets:

A ∩ B = {x | x ∈ A and x ∈ B}

Properties

• Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A

• Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)

Identities

• A ∪ ∅ = A

• A ∩ ∅ = ∅

7. Set Difference and Complement

Difference

A \ B = {x ∈ A | x ∉ B}

Complement

The set of all elements not in A within a given universal set U:

Aᶜ = U \ A

8. De Morgan’s Laws

(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

These laws relate complements to unions and intersections.

9. Power Sets and Sets of Sets

Power Set

The set of all subsets of a set A:

𝒫(A)

A = {0, 1} ⇒ 𝒫(A) = {∅, {0}, {1}, {0,1}}

Sets of Sets

Sets can contain other sets as elements.

10. Russell’s Paradox

Ω = {x | x ∉ x}

If Ω ∈ Ω, then Ω ∉ Ω

If Ω ∉ Ω, then Ω ∈ Ω

Leads to a paradox. 'Solved' using axiomatic set theory with strict rules (axioms).

See article on Sets — i.e., NOT solved!

11. Practical Applications

Set theory is used in:

• Mathematical logic

• Computer science

• Probability

• Database systems

• Defining logical statements and predicates

Summary

• Set = collection of elements

• Notation: curly brackets {}, ∈, ∉, |A|

• Set Builder: {x ∈ A | condition}

• Empty set: ∅, subset of all sets

• Union / Intersection: ∪, ∩

• Difference / Complement: \, ᶜ

• Equal sets: same elements regardless of order or repetition

• Power Set: all subsets

• Russell’s Paradox: motivates axiomatic theory

  • Next is Mathematics - Part 3: Complexity Theory

  • Completed with On Sets

  • Top is Mathematics - My Research

Final Grade and Comments

• ✅ Graded assignment by rdd13r on July 12th 2025. A