126 lines
3.4 KiB
Markdown
126 lines
3.4 KiB
Markdown
---
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tags:
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- set-theory
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---
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## Set theory
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Set theory is a sub-discipline of both mathematics and formal logic. In
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mathematics it is used as a universal framework for describing other
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mathematical theories. It is also utilised in computer science and linguistics.
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It is useful because it provides tools for modelling an extraordinary variety of
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structures.
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> Set theory and the theory of infinite sets was created by Georg Cantor
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> (1845-1918), a German mathematician.
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## Method of formalisation
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We can use the symbols of predicate logic to simplify and clarify natural
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language expression of set-theoretic principles. There are different ways to do
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this but we will use the standard quantifiers and:
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- variables $a,b,c,...$ to range over sets
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- variables $x,y,z$ to range over ordinary objects as well as sets.
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More generally we will use capital Latin letters ($A, B, ...$) to denote some
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specific set, i.e not a generalised/quantified notion of a set.
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### Example
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'Everything is a member of some set or another:
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$$ \forall x \exists a (x\in a) $$
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## What are sets?
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A set is a collection of objects. In mathematics the objects are mathematical
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objects.
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A **finite set:**
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$$ BG = { \textsf{Barry, Maurice, Robin}} $$
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An **infinite set:**
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$$ I = {1, 2, 3, 4, ...} $$
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> When we use braces to indicate the members of a set we are providing a **list
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> description** of the set.
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## Set membership
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If a set S is a collection of objects, to say that object x is a member of S is
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just to say that x is one of those objects.
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We might also express this in natural language as:
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- the object x is an element of the set S
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- the object x belongs to S
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- the set S contains the object x
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Formally, we use epsilon to express set membership:
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$$ x \in A $$
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This asserts that x is a member of the set A.
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The negation of this proposition is expressed:
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$$ x \notin A $$
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This asserts that x is not a member of the set A.
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### Subsets
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> Set A is a subset of set B if every member of A is also a member of B.
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For example the set of women is a subset of the set of humans because every
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woman is a human. We express subset relations like so:
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$$ A \subseteq B $$
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This asserts that set A is a subset of set B.
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The negation of this proposition is expressed:
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$$ A \not\subset B $$
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We must not confuse the relation of being a subset with being a member. Jane is
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a member of the set of women but Jane is not a subset of the set of women since
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Jane is not herself a set, she is an object/individual member.
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There is also the notion of a **proper subset.**
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> If subset _A_ of _B_ is a proper subset of _B_ then _B_ contains some elements
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> that are not in _A_.
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In other words, if B contains objects other than/ in addition to A.
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$$ A \subset B $$
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This asserts that set A is a proper subset of set B.
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For example, the set of women is a proper subset of the set of humans because
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the set of humans also includes the set of men. If there were only women and no
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men, then the set of women would be a subset of the set of humans.
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### Supersets
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If A is a subset of B then we say that B is a **superset** of A. Being a
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superset, B contains every object of A and may also contain other objects in
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addition to A. This is just a different way of asserting that A is a proper
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subset of B.
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$$ B \supseteq A $$
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This asserts B is a superset of A. The negation:
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$$ B \not\supset A $$
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This asserts that B is not a superset of A.
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## Resources
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[Set symbols](symbols.html)
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