A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set. The study of sets and their properties is the object of set theory.
Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.
Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.
Symbols used to operate on sets include (which means "and" or intersection), and (which means "or" or union). The symbol is used to denote the set containing no elements, called the empty set.
The notation , where and are arbitrary sets, is used to denote the set of maps from to , For example, an element of would be a map from the natural numbers to the set . Call such a function , then , , etc., are elements of , so call them , , etc. This now looks like a sequence of elements of , so sequences are really just functions from to.This notation is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.
Sequences and Tuples
A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
Sequences are to calculus what at calculator is to a scientist. There are many ways to introduce sequences. Here we will follow a somewhat unorthodox way. Indeed, consider a scientist doing an experiment; he is collecting data, let us say, every day. So, put 
to be the data collected the first day, 
be the data collected the second day, and so on.... and 
is the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic: there is an order on the number, that is, we naturally have the first number, the second number, and so on.... A sequence is by definition a set of real numbers with this natural order. We wil use the notation
,
to describe the sequence of numbers where is the nth number.
to be the data collected the first day, be the data collected the second day, and so on.... and is the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic: there is an order on the number, that is, we naturally have the first number, the second number, and so on.... A sequence is by definition a set of real numbers with this natural order. We wil use the notation , to describe the sequence of numbers where
is the nth number. A tuple is an ordered list of elements. In set theory, an (ordered) n-tuple is a sequence (or ordered list) of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2,7,4,1,7) denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets "[ ]" or angle brackets "
". Braces "{}" are almost never used for tuples, as they are the standard notation for sets.
". Braces "{}" are almost never used for tuples, as they are the standard notation for sets.Tuples are often used to describe other mathematical objects. In algebra, for example, a ring is commonly defined as a 3-tuple 
, where E is some set, and " + ", and "
" are functions mapping the Cartesian product 
to E with specific properties. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra.
, where E is some set, and " + ", and "" are functions mapping the Cartesian product to E with specific properties. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra.Functions and Relations
A function, in a mathematical sense, expresses the idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of the function. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which assigns to every real number the real number with twice its value. In this case, it is written that f(5) = 10.
include maps between algebraic structures like groups and maps between geometric objects like manifolds. In the abstract set-theoretic approach, a function is a relation between the domain and the codomain that associates each element in the domain with exactly one element in the codomain. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3.
There are many ways to describe or represent functions: by a formula, by an algorithm that computes it, by a plot or a graph. A table of values is a common way to specify a function in statistics, physics, chemistry, and other sciences. A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm.
Functions with numerical outputs may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the composition of functions.
A relation is any subset of a Cartesian product. For instance, a subset of 
, called a "binary relation from 
to 
," is a collection of ordered pairs 
with first components from 
and second components from 
, and, in particular, a subset of 
is called a "relation on 
." For a binary relation 
, one often writes 
to mean that 
is in 
, called a "binary relation from 
to 
," is a collection of ordered pairs 
with first components from 
and second components from 
, and, in particular, a subset of 
is called a "relation on 
." For a binary relation 
, one often writes 
to mean that 
is in 
Graphs
A graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
The edges may be directed (asymmetric) or undirected (symmetric). For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. On the other hand, if the vertices represent people at a party, and there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowing of someone is not necessarily a symmetric relation (that is, one person knowing of another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). This latter type of graph is called a directed graph and the edges are called directed edges or arcs; in contrast, a graph where the edges are not directed is called undirected.
Vertices are also called nodes or points, and edges are also called lines. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by James Joseph Sylvester in 1878.
Strings and Language
A string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and/or the length changed, or it may be fixed (after creation). A string is generally understood as a data type and is often implemented as a byte (or word) array that stores a sequence of elements, typically characters, using some character encoding. A string may also denote more general array data types and/or other sequential data types and structures; terms such as byte string, or more general, string of datatype, or datatype-string, are sometimes used to denote strings in which the stored data does not (necessarily) represent text.
Depending on programming language and/or precise datatype used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined max length, or it may employ dynamic allocation to allow it to hold chronologically variable number of elements. When a string appears literally in source code, it is known as a string literal and has a representation that denotes it as such. Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication. The scientific study of language in any of its senses is called linguistics.
The word "language" has two meanings: language as a general concept, and "a language" (a specific linguistic system, e.g. "French"). Languages other than English often have two separate words for these distinct concepts. French for example uses the word langage for language as a concept and langue as the specific instance of language.
Boolean Logic
Boolean logic is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of conjunction x∧y, disjunction x∨y, and negation ¬x. The Boolean operations are these and all other operations that can be built from these, such as x∧(y∨z).
The laws of Boolean algebra can be defined axiomatically as certain equations called axioms together with their logical consequences called theorems, or semantically as those equations that are true for every possible assignment of 0 or 1 to their variables. The axiomatic approach is sound and complete in the sense that it proves respectively neither more nor fewer laws than the semantic approach.
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