 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/2/c/ii/solution

Check each partial order axiom. For every

(a, b)

a \leq a

and

b \geq b

, so

S

(a, b) S (c, d)

and

(c, d) S (a, b)

, the first coordinates satisfy

a \leq c \leq a

and the second coordinates satisfy

b \geq d \geq b

. Therefore

a = c

and

b = d

, which proves

S

Finally, if

(a, b) S (c, d)

and

(c, d) S (e, f)

, then

a \leq c \leq e

and

b \geq d \geq f

. Hence

(a, b) S (e, f)

, making

S

a transitive relation. All three axioms hold, so $S$ is a partial order. It is a product order, with the usual order on the first coordinate and the reversed order on the second.

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