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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.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
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 uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals ...
Fully written out, this means that for any two pairs of elements > and > one must have > or >. The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form ( ℓ i , ℓ i + 1 ) {\displaystyle (\ell _{i},\ell _{i}+1)} , is precisely the semiorders .
Christmas can already be stressful enough. The last thing you want is anxiety about your pet's health. Here's how to cat-proof your home this holiday.
Weight machines get a bad rap, but are perfect for beginners to boost form and target muscles safely. Plus, a full-body workout programmed by a pro.
AAPL Market Cap data by YCharts. Other noteworthy examples include selling out of oil and gas stocks during the downturn of 2020. In the last four years, the energy sector is up 129%.
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