Sets:
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.
The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.
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.
is an element of a set 
. The study of sets and their properties is the object of set theory. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.
Sequences and Tuples
Sequence:
In mathematics, 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.
An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is bounded, however.
The two main types of series/sequences are arithmetic and geometric. Some sequences are neither of these. It’s important to be able to identify what type of sequence is being dealt with. And arithmetic series is one where each term is equal the one before it plus some number. For example: 5, 10, 15, 20, … Each term in this sequence equals the term before it with 5 added on. I n contrast, a geometric sequence is one where each term equals the one before it multiplied by a certain value. An example would be 3, 6, 12, 24, 48, … Each term is equal to the prior one multiplied by 2. Some sequences are neither arithmetic nor geometric. An example would be 1, 2, 3, 2, 1, 2, 3, 2, 1, …The terms in this sequence all differ by 1, but sometimes 1 is being added and other times it is being subtracted, so the sequence is not arithmetic. Also, there is no common value being multiplied by one term to get the next, so the sequence cannot be geometric, either. Arithmetic sequences grow very slowly in comparison with geometric sequences.
Tuple:
In mathematics and computer science, 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.
Tuples are often used to describe other mathematical objects. In algebra, for example, a ring is commonly defined as a 3-tuple (E,+,X), where E is some set, and " + ", and "X" are functions mapping the Cartesian product ExE 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.
A tuple is a sequence of values. The values can be any type, and they are indexed by integers, so in that respect tuples are a lot like lists. The important difference is that tuples are immutable.
A tuple (or n-tuple) is a fixed size collection of elements. Pairs, triples, quadruples etc. are tuples. In a programming language, a tuple is a data object containing other objects as elements. These element objects may be of different types. Tuples are convenient in many circumstances. For instance, tuples make it easy to define functions that return more than one value. Some programming languages, such as ML, Python and Haskell, have built-in tuple constructs. Unfortunately C++ does not. To compensate for this "deficiency", the Boost Tuple Library implements a tuple construct using templates.
Functions and Relations
Function:
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.
In addition to elementary functions on numbers, functions 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.
Both the domain and the range in the picture are the set of real numbers between −1 and 1.5.
Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argument and is often represented by the letter x or, if the input is a particular time, by the letter t. The symbol for the output is called the dependent variable or value and is often represented by the letter y. The function itself is most often called f, and thus the notation y = f(x) indicates that a function named f has an input named x and an output named y.
The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image or range of the function. The image is often a subset of some larger set, called the codomain of a function. Thus, for example, the function f(x) = x2 could take as its domain the set of all real numbers, as its image the set of all non-negative real numbers, and as its codomain the set of all real numbers. In that case, we would describe f as a real-valued function of a real variable. Sometimes, especially in computer science, the term "range" refers to the codomain rather than the image, so care needs to be taken when using the word.
Relation:
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.
An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (and not with any integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.
Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science.
A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain Aj of the relation.
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.
A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and G is called its graph.
Graphs:
In mathematics, 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.[1]
In a mathematician's terminology, a graph is a collection of points and lines connecting some (possibly empty) subset of them. The points of a graph are most commonly known as graph vertices, but may also be called "nodes" or simply "points." Similarly, the lines connecting the vertices of a graph are most commonly known as graph edges, but may also be called "arcs" or "lines."
The study of graphs is known as graph theory, and was first systematically investigated by D. König in the 1930s (Gardner 1984, p.)
Strings and Languages
String:
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet.
In computer programming, 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 allocatedmax 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.
String theory is a developing theory in particle physics that attempts to reconcile quantum mechanics and general relativity.[1] It is a contender for the theory of everything (TOE), a manner of describing the known fundamental forces and matter in a mathematically complete system. The theory has yet to make testable experimental predictions, which a theory must do in order to be considered a part of science.
String theory mainly posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines ("strings"). The earliest string model, the bosonic string, incorporated only bosons, although this view developed to the superstring theory, which posits that a connection (a "supersymmetry") exists between bosons and fermions. String theories also require the existence of several extra, unobservable, dimensions to the universe, in addition to the usual four spacetime dimensions.
The theory has its origins in the dual resonance model (1969). Since that time, the term string theory has developed to incorporate any of a group of related superstring theories. Five major string theories were formulated. The main differences among them were the number of dimensions in which the strings developed and their characteristics; all of them appeared to be correct, however. In the mid 1990s a unification of all previous superstring theories, called M-theory, was proposed, which asserted that strings are really 1-dimensional slices of a 2-dimensional membrane vibrating in 11-dimensional space.
As a result of the many properties and principles shared by these approaches (such as the holographic principle), their mutual logical consistency, and the fact that some easily include the standard model of particle physics, some mathematical physicists (e.g. Witten, Maldacena and Susskind) believe that string theory is a step towards the correct fundamental description of nature.[2][3][4][5][unreliable source?] Nevertheless, other prominent physicists (e.g. Feynman and Glashow) have criticized string theory for not providing any quantitative experimental predictions.[6][7]
Language:
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 approximately 3000–6000 languages that are spoken by humans today are the most salient examples, but natural languages can also be based on visual rather than auditive stimuli, for example in sign languages and written language. Codes and other kinds of artificially constructed communication systems such as those used for computer programming can also be called languages. A language in this sense is a system of signs for encoding and decoding information. The English word derives from Latin lingua, "language, tongue." This metaphoric relation between language and the tongue exists in many languages and testifies to the historical prominence of spoken languages.[1] When used as a general concept, "language" refers to the cognitive faculty that enables humans to learn and use systems of complex communication.
Boolean Logic:
Boolean logic, developed by George Boole (1815-1864), is often used to refine the determination of system status or to set or clear specific bits. Boolean logic is simply a way of comparing individual bits. It uses what are called operators to determine how the bits are compared. They simulate the gates that you will see in the hardware section you will read shortly.
Digital logic is the application of the Boolean algebra of 0 and 1 to electronic hardware consisting of logic gates connected to form a circuit diagram. Each gate implements a Boolean operation, and is depicted schematically by a shape indicating the operation. The shapes associated with the gates for conjunction (AND-gates), disjunction (OR-gates), and complement (inverters) are as follows.[13]
The lines on the left of each gate represent input wires or ports. The value of the input is represented by a voltage on the lead. For so-called "active-high" logic 0 is represented by a voltage close to zero or "ground" while 1 is represented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gate represents the output port, which normally follows the same voltage conventions as the input ports.Complement is implemented with an inverter gate. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port.
The original application for Boolean operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.
Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies). But not is synonymous with and not. When used to combine situational assertions such as "the block is on the table" and "cats drink milk," which naively are either true or false, the meanings of these logical connectives often have the meaning of their logical counterparts. However with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for example the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, since and usually means and then in such cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as in get dressed and go to school. Disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns such as tea and milk generally describe aggregation as with set union while tea or milk is a choice. However context can reverse these senses, as in your choices are coffee and tea which usually means the same as your choices are coffee or tea (alternatives). Double negation as in "I don't not like milk" rarely mean literally "I do like milk" but rather convey some sort of hedging, as though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and although P necessarily implies "not not P" the converse is suspect in English, much as with intuitionistic logic. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.
Boolean operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector of n identical binary gates are used to combine two bit vectors each of n bits, the individual bit operations can be understood collectively as a single operation on values from a Boolean algebra with 2n elements.
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