Operation (mathematics) in the context of "Algebraic structure"

In mathematics, exponentiation, denoted b, is an operation involving two numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b is the product of multiplying n bases:$${\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}$$In particular, ${\displaystyle b^{1}=b}$.

The exponent is usually shown as a superscript to the right of the base as b or in computer code as b^n. This binary operation is often read as "b to the power n"; it may also be referred to as "b raised to the nth power", "the nth power of b", or, most briefly, "b to the n".

In mathematics, a symbolic language is a language that uses characters or symbols to represent concepts, such as mathematical operations, expressions, and statements, and the entities or operands on which the operations are performed.

In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.

An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of the vector space (called vectors).

In mathematics, an expression is an arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

Expressions are commonly distinguished from formulas: expressions usually denote mathematical objects, whereas formulas are statements about mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, ${\displaystyle 8x-5}$ and ${\displaystyle 3}$ are both expressions, while the inequality ${\displaystyle 8x-5\geq 3}$ is a formula. However, formulas are often considered as expressions that can be evaluated to the Boolean values true or false.

In mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.

An algebraic operation on a set ⁠${\displaystyle S}$⁠ may be defined more formally as a function that maps to ⁠${\displaystyle S}$⁠ the tuples of a given length of elements of ⁠${\displaystyle S}$⁠. The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five".

Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups.

Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces numerical variables (quantities without fixed values).

This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.

A partial list of possible structures is measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, graphs, events, differential structures, categories, setoids, and equivalence relations.

In logic, mathematics, and computer science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.

In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition ${\displaystyle P}$ to another proposition "not ${\displaystyle P}$", written ${\displaystyle \neg P}$, ${\displaystyle {\mathord {\sim }}P}$, ${\displaystyle P^{\prime }}$ or ${\displaystyle {\overline {P}}}$. It is interpreted intuitively as being true when ${\displaystyle P}$ is false, and false when ${\displaystyle P}$ is true. For example, if ${\displaystyle P}$ is "The dog runs", then "not ${\displaystyle P}$" is "The dog does not run". An operand of a negation is called a negand or negatum.

Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition ${\displaystyle P}$ is the proposition whose proofs are the refutations of ${\displaystyle P}$.