## Mathematical Notions And Terminology

As in any mathematical subject , we begin with a discussion of the basic mathematical objects, tools, and notation that we expect to use.

## SET

A set is a group of objects represented as a unit. Sets may contain any type of object, including numbers, symbols, and even other sets.

- The objects in a set are called its
or**elements**.**members** - For sets, we use a type of picture called a
**Venn diagram** - The set with 0 members is called the
**empty set**

## SEQUENCES AND TUPLES

A sequence of objects is a list of these objects in *some order*. As with sets, sequences may be *finite or infinite*. Finite sequences often are called *tuples*

- We usually designate a sequence by writing the list within
**parentheses**. - In a
**set, the order doesn’t matter**, but in a sequence it does **Repetition does matter in a sequence**, but it does not matter in a set

## FUNCTIONS AND RELATIONS

A function is an object that sets up input-output relationship. A function takes an input and produces an output.

- A function is called a
if*mapping**f(a)*= b, we say that*f*maps*a*to*b*. - The set of possible inputs to the function is called its
.*domain* - The outputs of a function come from a set called its
.*range* - The notation for saying that
*f is a function with domain D and range R*is*f*: D –> R - A function may not necessarily use all the elements of the specified range.
- A function that does use all the elements of the range is said to be
the range.*onto*

## GRAPHS

An * undirected graph*, or simply a

*, is a set of points with lines connecting some of the points.*

**graph**- The points are called
or*nodes*, and the lines are called*vertices*.*edges* - The number of edges at a particular node is the
of that node.*degree* **Graphs**frequently are used to represent data.- A
**path**in a graph is a sequence of nodes connected by edges. - A
**simple path**is a path that doesn’t repeat any nodes. - A
**graph is connected**if every two nodes have a path between them. - A
**path is a cycle**if it starts and ends in the same node. - A
**simple cycle**is one that contains at least three nodes and repeats only the first and last nodes. - A graph is a
**tree**if it is connected and hs no simple cycles

## Mathematical Notions And Terminology Summary

Alphabet | A finite set of objects called symbols |

Argument | An input to a function |

Binary operation | A relation whose domain is a set of pairs |

Cartesian product | An operation on sets forming a set of all tuples of elements from respective sets |

Cycle | A path that starts and ends in the same node |

Domain | The set of possible inputs to a function |

Edge | A line in a graph |

Element | An object in a set |

Empty String | The string of length zero |

Function | An operation that translates inputs inot outputs |

Graph | A collection of points and lines connecting some pairs of points |

Language | A set of strings |

Member | An object in a graph |

Node | A point in a graph |

Pair | A list of two elements, also called a 2-tuple |

Path | A sequence of nodes in a graph connected by edges |

Property | A predicate |

Predicate | A function whose range is { TRUE, FALSE } |

Range | The set from which outputs of a function are drawn |

Vertex | A point in a graph |