Orderings & Lattices · step 4 of 13

Upper & lower bounds, supremum & infimum

An upper bound of a subset S is an element above everything in S; the supremum is the least such bound, if one exists. Infimum is the dual notion. The key subtleties are that bounds need not be members of S, that a subset can have many bounds but at most one supremum, and that a supremum may fail to exist even when bounds do.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

List all upper boundsFind every element above every member of S. Missing one can make you name the wrong supremum.
Take the least of themThe supremum is the minimum of the set of upper bounds — so it must be comparable to and below all the others.
Repeat downward for the infimumLower bounds are elements below everything in S; the infimum is the greatest of them.
Check existence honestlyIf the upper bounds have no least element, sup S does not exist. Say so — inventing one is the classic error.

Worked example

In the divisibility poset on divisors of 36, find sup{4, 6} and inf{4, 6}.

Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Upper bounds of {4,6} are common multiples in the set: 12 and 36.
The least of these under divisibility is 12, since 12 ∣ 36. So sup = 12 = lcm(4,6).
Lower bounds are common divisors: 1 and 2. The greatest is 2 = gcd(4,6).

Answer. sup{4,6} = 12 and inf{4,6} = 2 — in a divisibility poset, sup is lcm and inf is gcd.

Where marks get dropped

These are the specific errors that cost credit on upper & lower bounds, supremum & infimum questions — QED's rubric penalises each of them separately.

Choosing any upper bound instead of the least. 36 bounds {4,6} but is not the supremum.
Requiring the supremum to belong to S. sup{4,6} = 12 is not in {4,6}, and that is perfectly normal.
Assuming existence. In ℚ the set {x : x² < 2} is bounded above but has no rational supremum — the completeness of ℝ is precisely the axiom that fixes this.

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Upper & lower bounds, supremum & infimum — frequently asked questions

What is the difference between maximum and supremum?

A maximum must belong to the set; a supremum need not. For (0,1) the supremum is 1 but there is no maximum.

Why is sup unique?

It is the least element of the set of upper bounds, and least elements are unique by antisymmetry.

What does completeness mean?

A poset is complete if every subset has a supremum. ℝ is complete while ℚ is not, and that single difference is what makes analysis work over ℝ.

The rest of Orderings & Lattices

Partial orders, Hasse diagrams, bounds, lattices. Each subtopic below has its own method, worked example and mark-losing traps.

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