Scalar (mathematics)In mathematics, more specifically in linear algebra, a scalar is an element of a field which is used to define a vector space through the operation of scalar multiplication: a vector (denoted v) multiplied by a scalar (denoted a) produces another vector (av). Real numbers and complex numbers may be used as scalars in real and complex vector spaces, respectively.
CodomainIn mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in the notation f : X → Y. The term range is sometimes ambiguously used to refer to either the codomain or the image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph.
ConvolutionIntegral expressing the amount of overlap of one function as it is shifted over anotherVisual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f{\displaystyle f}, and assuming the height of f{\displaystyle f} is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point.
Algebraic structureIn mathematics, an algebraic structure or algebraic system consists of a nonempty setA (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.
ArityIn logic, mathematics, and computer science, arity () 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.
Domain of a functionIn mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom(f){\displaystyle \operatorname {dom} (f)} or domf{\displaystyle \operatorname {dom} f}, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".
Function (mathematics)Association of one output to each inputIn mathematics, a function from a setX to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Associative propertyIn mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a validrule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.
Commutative propertyIn mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings.
Distributive propertyIn mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x⋅(y+z)=x⋅y+x⋅z{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} is always true in elementary algebra. For example, in elementary arithmetic, one has 2⋅(1+3)=(2⋅1)+(2⋅3).{\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).} Therefore, one would say that multiplicationdistributes over addition.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Finitary relationIn mathematics, a finitary relation over a sequence of sets X1, ..., Xn is a subset of the Cartesian productX1 × ... × Xn; that is, it is a set of n-tuples(x1, ..., xn), each being a sequence of elements xi in the corresponding Xi. Typically, the relation describes a possible connection between the elements of an n-tuple.*As an Amazon Associate I earn from qualifying purchases.