# Definition:Set Theory

## Definition

**Set Theory** is the branch of mathematics which studies sets.

There are several "versions" of set theory, all of which share the same basic ideas but whose foundations are completely different.

### Naive set theory

**Naïve set theory**, in contrast with axiomatic set theory, is an approach to set theory which assumes the existence of a universal set, despite the fact that such an assumption leads to paradoxes.

A popular alternative (and inaccurate) definition describes this as a

*non-formalized definition of set theory which describes sets and the relations between them using natural language.*

However, the discipline is founded upon quite as rigid a set of axioms, namely, those of propositional and predicate logic.

### Axiomatic set theory

**Axiomatic set theory** is a system of set theory which differs from so-called naive set theory in that the sets which are allowed to be generated are strictly constrained by the axioms.

### Pure set theory

**Pure set theory** is a system of set theory in which all elements of sets are themselves sets.

## Examples

### Unions and Intersections $1$

Let:

\(\ds V_1\) | \(=\) | \(\ds \set {v_1, v_3, v_4}\) | ||||||||||||

\(\ds V_2\) | \(=\) | \(\ds \set {v_2, v_5}\) | ||||||||||||

\(\ds V_3\) | \(=\) | \(\ds \set {v_1, v_3}\) |

Then:

\(\ds V_1 \cup V_2\) | \(=\) | \(\ds \set {v_1, v_2, v_3, v_4, v_5}\) | ||||||||||||

\(\ds V_1 \cup V_3\) | \(=\) | \(\ds \set {v_1, v_3, v_4}\) | ||||||||||||

\(\ds V_2 \cup V_3\) | \(=\) | \(\ds \set {v_1, v_2, v_3, v_5}\) | ||||||||||||

\(\ds V_1 \cap V_2\) | \(=\) | \(\ds \O\) | ||||||||||||

\(\ds V_1 \cap V_3\) | \(=\) | \(\ds \set {v_1, v_3}\) | ||||||||||||

\(\ds V_2 \cap V_3\) | \(=\) | \(\ds \O\) |

Thus:

### Unions and Intersections $2$

Let:

\(\ds A\) | \(=\) | \(\ds \set {1, 2}\) | ||||||||||||

\(\ds B\) | \(=\) | \(\ds \set {1, \set 2}\) | ||||||||||||

\(\ds C\) | \(=\) | \(\ds \set {\set 1, \set 2}\) | ||||||||||||

\(\ds D\) | \(=\) | \(\ds \set {\set 1, \set 2, \set {1, 2} }\) |

Then:

\(\ds A \cap B\) | \(=\) | \(\ds \set 1\) | ||||||||||||

\(\ds \paren {B \cap D} \cup A\) | \(=\) | \(\ds \set {1, 2, \set 2}\) | ||||||||||||

\(\ds \paren {A \cap B} \cup D\) | \(=\) | \(\ds \set {1, \set 1, \set 2, \set {1, 2} }\) | ||||||||||||

\(\ds \paren {A \cap B} \cup \paren {C \cap D}\) | \(=\) | \(\ds \set {1, \set 1, \set 2}\) |

### Equations $A \cup \paren {X \cap B} = C$, $\paren {A \cup X} \cap B = D$

Let $A, B, C, D$ be subsets of a set $S$.

Let there exist $X \subseteq S$ such that:

- $A \cup \paren {X \cap B} = C$
- $\paren {A \cup X} \cap B = D$

Then:

- $A \cap B \subseteq D \subseteq B$

and:

- $A \cup D = C$

$\blacksquare$

## Also see

- Results about
**set theory**can be found here.

## Historical Note

Set theory arose from an attempt to comprehend the question: "What is a number?"

The main initial development of the subject was in fact not directly generated as a result of trying to answer this question, but as a result of Georg Cantor's work around $1870$ to understand the nature of infinite series and related subjects.

As a result of this he began to consider the nature of infinite collections of general object, not just numbers.

*Cantor ....is usually considered the founder of set theory as a mathematical discipline ...*- -- Patrick Suppes:
*Axiomatic Set Theory*(1960, 2nd ed. 1972)

- -- Patrick Suppes:

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (next): Preface

*... General set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it.*

- 1965: A.M. Arthurs:
*Probability Theory*... (next): Chapter $1$: Set Theory: $1.1$: Introduction

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems

*It would be completely out of the question at this stage ... to attempt an axiomatisation of such topics ...*

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?

*In set theory, there is really only one fundamental notion:*

- 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson