 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability/2/c/i/solution

(A, ⪯)

, the assertion

m = lub (S)

means both

(\forall s \in S, s ⪯ m) and (\forall u \in A, [\forall s \in S, s ⪯ u] \Rightarrow m ⪯ u) .

The first condition makes

m

an upper bound; the second makes it the least upper bound. Both conditions are necessary, and antisymmetry makes such an element unique.

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