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In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ...
In mathematics, the real numbers may be described informally in several different ways. ...
Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found in that article. In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
Infima of real numbers
In analysis the infimum or greatest lower bound of a set S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because S is not bounded below), then we define inf(S) = −∞. If S is empty, we define inf(S) = ∞ (see extended real number line). Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...
Please refer to Real vs. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...
An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers).
Examples:
    If a set has a smallest element, as in the first example, then the smallest element is the infimum for the set. As the last three examples shows, the infimum of a set does not have to belong to the set.
The notions of infimum and supremum are dual in the sense that In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
 , where  .
In general, in order to show that inf(S) ≥ A, one only has to show that x ≥ A for all x in S. Showing that inf(S) ≤ A is a bit harder: for any ε > 0, you have to exhibit an element x in S with x ≤ A + ε (of course, if you can find an element x in S with x ≤ A, you are done right away).
See also: limit inferior. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ...
Infima in partially ordered sets The definition of infima easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory. In this context, especially in lattice theory, greatest lower bounds are also called meets. In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
Formally, the infimum of a subset S of a partially ordered set (P, ≤) is an element l of P such that
- l ≤ x for all x in S, and
- for any p in P such that p ≤ x for all x in S it holds that p ≤ l.
Any element with these properties is necessarily unique, but in general no such element needs to exist. Consequently, orders for which certain infima are known to exist become especially interesting. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ...
The dual concept of infimum is given by the notion of a supremum or least upper bound. By the duality principle of order theory, every statement about suprema is thus readily transformed into a statement about infima. For this reason, all further results, details, and examples can be taken from the article on suprema. In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
Least upper bound property See the article on the least upper bound property. In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...
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