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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {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.
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.
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Given a set X, a relation R over X is a set of ordered pairs of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X}. [2] [10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy. [7] [8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.
It follows that, two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d. Alternatively, an ordered pair can be formally thought of as a set {a,b} with a total order. (The notation (a, b) is also used to denote an open interval on the real number line, but the context should make it clear which meaning is intended.
An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {♠, ♥, ♦, ♣} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
In mathematics, a pointed set [1] [2] (also based set [1] or rooted set [3]) is an ordered pair (,) where is a set and is an element of called the base point, [2] also spelled basepoint. [ 4 ] : 10–11