Definition:Set Theory

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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:

$V_1$ and $V_2$ are disjoint
$V_2$ and $V_3$ are disjoint.


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.


In $1873$, Cantor discovered that the set of algebraic reals is countable.

Soon after that, he discovered that the set of all real numbers is uncountable.

The concepts of equipotent sets, order isomorphic structures, cardinals and ordinals are all due to Cantor.


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


Sources

... 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.
It would be completely out of the question at this stage ... to attempt an axiomatisation of such topics ...
In set theory, there is really only one fundamental notion:
The ability to regard any collection of objects as a single entity (i.e. as a set).
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