Set Theory

Phrase	Meaning	Definition
x∈M	x is an element of M
A⊆B	A ist a subset of B
A=B	The sets are equal
A⊂B	A is an actual subset of B
A\B	A without elements of B
0	empty set
A∩B	intersection of A and B
A∪B	union of A and B
2^M=P(M)	Power set of M: set of all subsets of M.
|A|	Cardinality of A (usually number of elements)

Relations

A Relation R relating A to B is a set of tuples (a,b) where a∈A, b∈B (which tuples these are defines a specific relation).

Functions

A Function f : A -> B assigns, for some elements of A, an element of the set A (that set is called the domain) to an element of the set B (that set is called the codomain). A Function is a Relation such that if a is related to b, then a is not related to any other element the same time.

Partial Functions

A Function f does not have to assign a value for each element of A (the domain). If it doesn't, it's called a Partial Function.

Total Functions

A Function f does not have to assign a value for each element of A. If it does, it's called a Total Function.

Range

The range is the set of values that result from applying f (pointwise) to all elements of the domain.

The range is not (necessarily) the codomain.

Image

The Image of a Function is the set of values that result from applying f (pointwise) to some specified set (not necessarily the entire domain).

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .