Injective function in the context of Embedding (mathematics)

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.

In the context of digital signal processing (DSP), a digital signal is a discrete time, quantized amplitude signal. In other words, it is a sampled signal consisting of samples that take on values from a discrete set (a countable set that can be mapped one-to-one to a subset of integers). If that discrete set is finite, the discrete values can be represented with digital words of a finite width. Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words.

The process of analog-to-digital conversion produces a digital signal. The conversion process can be thought of as occurring in two steps:

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

When some object ${\displaystyle X}$ is said to be embedded in another object ${\displaystyle Y}$, the embedding is given by some injective and structure-preserving map ${\displaystyle f:X\rightarrow Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ${\displaystyle X}$ and ${\displaystyle Y}$ are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal). Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".

In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

In computer science, a perfect hash function h for a set S is a hash function that maps distinct elements in S to a set of m integers, with no collisions. In mathematical terms, it is an injective function.

Perfect hash functions may be used to implement a lookup table with constant worst-case access time. A perfect hash function can, as any hash function, be used to implement hash tables, with the advantage that no collision resolution has to be implemented. In addition, if the keys are not in the data and if it is known that queried keys will be valid, then the keys do not need to be stored in the lookup table, saving space.

In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image.

When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.

In mathematics, the disjoint union (or discriminated union) ${\displaystyle A\sqcup B}$ of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.

A disjoint union of an indexed family of sets ${\displaystyle (A_{i}:i\in I)}$ is a set ${\displaystyle A,}$ often denoted by ${\textstyle \bigsqcup _{i\in I}A_{i},}$ with an injection of each ${\displaystyle A_{i}}$ into ${\displaystyle A,}$ such that the images of these injections form a partition of ${\displaystyle A}$ (that is, each element of ${\displaystyle A}$ belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.