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  2. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.

  3. Completeness (order theory) - Wikipedia

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

    It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema. Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds.

  4. Order theory - Wikipedia

    en.wikipedia.org/wiki/Order_theory

    For example, 1 is the infimum of the positive integers as a subset of integers. For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is the smallest number that is divided by both of them, i.e. the least common multiple of the numbers.

  5. Limit inferior and limit superior - Wikipedia

    en.wikipedia.org/wiki/Limit_inferior_and_limit...

    In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...

  6. Least-upper-bound property - Wikipedia

    en.wikipedia.org/wiki/Least-upper-bound_property

    A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real ...

  7. Completeness of the real numbers - Wikipedia

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

    The least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded above) must have a least upper bound (or supremum) in the set of real numbers. The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers

  8. List of set identities and relations - Wikipedia

    en.wikipedia.org/wiki/List_of_set_identities_and...

    8.3.1 Counter-examples: ... is the join/supremum of and with respect to because: and , ... is the meet/infimum of and with respect to . Other inclusion ...

  9. Dedekind–MacNeille completion - Wikipedia

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

    A complete lattice is a lattice in which every subset of elements of L has an infimum and supremum; this generalizes the analogous properties of the real numbers. An order-embedding is a function that maps distinct elements of S to distinct elements of L such that each pair of elements in S has the same ordering in L as they do in S.