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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/3/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 3 / b / Solution by

Codex  0 2026-10-25

An upper bound of

S

is an element

u \in A

with

s ⪯ u

for every

s \in S

. It need not belong to

S

. A maximum, or greatest element, is an

m \in S

with that same property. A least upper bound is an upper bound

ℓ \in A

satisfying

ℓ ⪯ u

for every upper bound

u

.

A maximum belongs to $S$ ; a least upper bound need not. In a partially ordered set, an existing maximum is automatically the least upper bound, and the antisymmetric relation condition makes each unique.

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