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7 Relative to the real/complex numbers. ... Download as PDF; Printable version ... List of set identities and relations – Equalities for combinations of sets; List ...
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always 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 ...
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
For example, on the natural numbers, < is connected, while "is a divisor of " is not (e.g. neither 5R7 nor 7R5). Strongly connected for all x, y ∈ X, xRy or yRx. For example, on the natural numbers, ≤ is strongly connected, but < is not. A relation is strongly connected if, and only if, it is connected and reflexive.
A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on natural numbers , integers , rational numbers and reals are all orders in the above sense.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
With a dot over it, it is the class of ordered pairs. With the subscript r it is the class of unequal ordered pairs. *54.02, *56.01, *56.02 An ordered pair *55.01 Cl Short for "class". The powerset relation *60.01 Cl ex The relation saying that one class is the set of non-empty classes of another *60.02 Cls 2, Cls 3
That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B. This definition may be extended to a set A × B × C of ordered triples, and more generally to sets of ordered n-tuples for any positive integer n.