 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/1/b/ii/solution

p (x) \leq p (x)

proves reflexivity. Two opposite inequalities imply

p (x) = q (x)

for every real

x

x

To disprove total order, take

p (x) = x

and

q (x) = 0

. The inequality

p ⪯ q

fails at

x = 1

, whereas

q ⪯ p

fails at

x = - 1

. The order is partial but non-total.

 New to topics? Read the docs here!