Mathematical Notions And Terminology

Mathematical Notions And Terminology

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 elements or 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 mapping if 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 onto the range.

GRAPHS

An undirected graph, or simply a graph, is a set of points with lines connecting some of the points.

  • The points are called nodes or vertices, and the lines are called edges.
  • The number of edges at a particular node is the degree of that node.
  • 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

AlphabetA finite set of objects called symbols
ArgumentAn input to a function
Binary operationA relation whose domain is a set of pairs
Cartesian productAn operation on sets forming a set of all tuples of elements from respective sets
CycleA path that starts and ends in the same node
DomainThe set of possible inputs to a function
EdgeA line in a graph
ElementAn object in a set
Empty StringThe string of length zero
FunctionAn operation that translates inputs inot outputs
GraphA collection of points and lines connecting some pairs of points
LanguageA set of strings
MemberAn object in a graph
NodeA point in a graph
PairA list of two elements, also called a 2-tuple
PathA sequence of nodes in a graph connected by edges
PropertyA predicate
PredicateA function whose range is { TRUE, FALSE }
RangeThe set from which outputs of a function are drawn
VertexA point in a graph

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