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2. Completeness of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Completeness_of_the_real...

   In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number. Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction.

3. Construction of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Construction_of_the_real...

   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 ...

4. Ordered pair - Wikipedia

   en.wikipedia.org/wiki/Ordered_pair

   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.

5. Real number - Wikipedia

   en.wikipedia.org/wiki/Real_number

   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 ...

6. Dedekind–MacNeille completion - Wikipedia

   en.wikipedia.org/wiki/Dedekind–MacNeille...

   A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.

7. Completeness (order theory) - Wikipedia

   en.wikipedia.org/wiki/Completeness_(order_theory)

   A poset is a complete lattice if and only if it is a cpo and a join-semilattice. Indeed, for any subset X, the set of all finite suprema (joins) of X is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum of X. Thus every set has a supremum and by the above observation we have a complete lattice.

8. Your Cholesterol Could Be A Key Indicator Of Dementia. A ...

   www.aol.com/cholesterol-could-key-indicator...

   Typically, dementia is associated with classic symptoms like confusion and memory loss. But new research finds that there could be a less obvious risk factor out there: your cholesterol levels ...

9. Interval (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Interval_(mathematics)

   The notation [a, b] too is occasionally used for ordered pairs, especially in computer science. Some authors such as Yves Tillé use ]a, b[to denote the complement of the interval (a, b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.