 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-resit/3/b/solution

This is the pointwise order on a function space. Each of the three partial order axioms follows from the corresponding axiom in

B

f (x) ⪯_{B} f (x)

for every

x

, so

f ⪯_{F} f

f ⪯_{F} g

and

g ⪯_{F} f

, then

f (x) = g (x)

for every

x \in A

; equality of functions therefore gives

f = g

f ⪯_{F} g ⪯_{F} h

, then

f (x) ⪯_{B} g (x) ⪯_{B} h (x)

for every

x

, so

f ⪯_{F} h

Thus $(F, ⪯_{F})$ is a partially ordered set. The argument also covers an empty domain

A

, since then there is exactly one function

A \to B

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