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Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound. In , the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. In contrast, 0 is the number that is divided by all other numbers. Hence it is the greatest element of the order. Other frequent terms for the least and greatest elements is bottom and top or zero and unit. Least and greatest ...
Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds ...
In second-order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem. [7] There is a well-known joke about the three statements, and their relative amenability to intuition:
Other common names for the least element are bottom and zero (0). The dual notion, the empty lower bound, is the greatest element, top, or unit (1). Posets that have a bottom are sometimes called pointed, while posets with a top are called unital or topped. An order that has both a least and a greatest element is bounded.
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The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S {\displaystyle S} of a preordered set is an element of S {\displaystyle S} which is greater than or equal to any other element of S , {\displaystyle S,} and ...