Lower Bound, Greatest Lower Bound (GLB) -- Infimum

This article is about lower bounds in sets. If you’re looking for bounds in summations, see: summation (sigma) notation. Or, for integrals, see: integral bounds.

In simple terms, the lower bound of an ordered set is where the lower boundary is. In set theory and real analysis, it’s defined more strictly as: an element a where every other element in the set is greater than or equal to a. This sounds similar, but there’s a big difference: a lower bound doesn’t actually have to be in the set.

For example, in the set {2, 4, 6, 7, 8, 10}, the element 2 is a lower bound. However, any number less than 2 is also a lower bound.

If this sounds like an exercise in futility, an example should help to clear this up. Let’s say you’re performing an experiment where you’re measuring beagles’ weights. You weigh 6 beagles and get
{22 lbs, 25 lbs, 27 lbs, 27 lbs, 29 lbs, 30 lbs}.
You could say that 22 pounds is a lower bound, and that would be valid. However, it’s possible that some beagles weigh less (perhaps they are old and frail, the runt of the litter, or are ill). So you set a low bound at 19 pounds instead (you could have chosen 15 lbs, 21 pounds, or any other number less than 22 pounds). Both values—19 lbs and 22 lbs are valid lower bounds for this set.

Greatest Lower Bound

While there can be many lower bounds, there can be only one greatest lower bound (GLB or infimum).

Formally, the GLB is defined as follows (Burn, R. 2000, p.4):

When (i) l is a lower bound for a set A of real numbers, and (ii) every m, greater than l, is not a lower bound for A, then and only then, l is called the greatest lower bound of A, and denoted by inf A.

To find the GLB, first take the set of all lower bounds. In the beagle example above, the set (in pounds) is {… 20, 21, 22}. The greatest number here is 22 lbs, so that is the GLB.

Note though, that the GLB may not always exist.

References

Burn, R. (2000). Numbers and Functions: Steps into Analysis 2nd Edition, Cambridge University Press.

Rodgers, N. (2000), Learning to Reason: An Introduction to Logic, Sets, and Relations. Wiley-Interscience; 1 edition.
Beagle image: By Soccersmp – https://commons.wikimedia.org/wiki/File:Beagle_puppy_Cadet.jpg, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=76344103