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).