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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.
Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle. The number of different chord diagrams that may be given for a set of 2 n {\displaystyle 2n} cyclically ordered objects is the double factorial ( 2 n − 1 ) ! ! {\displaystyle (2n ...
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.
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 ...
The key thing that differentiates reindeer from caribou is the fact that a reindeer is domesticated and a caribou is wild. From a distance, you probably wouldn't be able to tell a reindeer or a ...
The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, and thus it is sometimes omitted.
Cladonia portentosa, also known as reindeer lichen or the cream cup lichen, [1] is a light-coloured, fruticose, cup lichen in the family Cladoniaceae. A similar-looking species, also known by the common name "reindeer lichen", is Cladonia rangiferina .
Reindeer were imported from Siberia in the late 19th century and from Norway in the early 1900s as semi-domesticated livestock in Alaska. [45] [46] Reindeer can interbreed with the native caribou subspecies, but they rarely do, and even then their offspring do not survive well in the wild. [47] [25]
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