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A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells ...
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 ...
This means that the n 2 ordered pairs (r, c) are all the pairs (i, j) with 1 ≤ i, j ≤ n, once each. The same is true of the ordered pairs (r, s) and the ordered pairs (c, s). The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.
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.
An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes , [ 5 ] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product .
The example at left is that of an orthogonal array with symbol set {1,2} and strength 2. Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns, namely (1,1), (2,1), (1,2) and (2,2), are all the possible ordered pairs of the two element set and each appears exactly once.
Cartesian plane with marked points (signed ordered pairs of coordinates). For any point, the abscissa is the first value (x coordinate), and the ordinate is the second value (y coordinate). In mathematics , the abscissa ( / æ b ˈ s ɪ s . ə / ; plural abscissae or abscissas ) and the ordinate are respectively the first and second coordinate ...
First, the enumerated set { (x, 1), (y, 2), (z, 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as {x, y} = 011 (2); x of S is located at the first from the right of this sequence and y is at the second from the right, and 1 in the sequence means the ...