Up toTwo mathematical objects a and b are called equal up to an equivalence relation R if a and b are related by R, that is, if aRb holds, that is, if the equivalence classes of a and b with respect to R are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, x is unique up to R means that all objects x under consideration are in the same equivalence class with respect to the relation R.
Class (set theory)In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox (see ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.
Ordered pairIn mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.
Field of fractionsIn abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of an integral domain is sometimes denoted by or , and the construction is sometimes also called the fraction field, field of quotients, or quotient field of .
Quotient (universal algebra)In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below. Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.
Symmetric relationA symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if: where the notation means that . If RT represents the converse of R, then R is symmetric if and only if R = RT. Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. "is equal to" (equality) (whereas "is less than" is not symmetric) "is comparable to", for elements of a partially ordered set ".
Well-defined expressionIn mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if takes real numbers as input, and if does not equal then is not well defined (and thus not a function).
Quotient moduleIn algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup).
Projection (set theory)In set theory, a projection is one of two closely related types of functions or operations, namely: A set-theoretic operation typified by the th projection map, written that takes an element of the Cartesian product to the value A function that sends an element to its equivalence class under a specified equivalence relation or, equivalently, a surjection from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image.
