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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/2/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 2 / a / Solution by

Codex  0 2026-10-25

For real-valued functions, use magnitude comparisons. The constants below are independent of

n

.

The statement

f (n) = O (g (n))

in Big O notation means that some

C > 0

N

satisfy

∣ f (n) ∣ \leq C ∣ g (n) ∣

for all

n \geq N

. The statement

f (n) = Ω (g (n))

in big omega notation means that some

c > 0

and

N

satisfy

∣ f (n) ∣ \geq c ∣ g (n) ∣

for all

n \geq N

.

Finally,

f (n) = Θ (g (n))

in big theta notation means that some

c, C > 0

and

N

satisfy

c ∣ g (n) ∣ \leq ∣ f (n) ∣ \leq C ∣ g (n) ∣ (n \geq N) .

Equivalently it is both an eventual upper bound and an eventual lower bound. These definitions ignore a finite initial segment; they do not assert pointwise inequalities with constant one.

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