Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Discrete mathematics / 3 / c / i

Table of contents
- Solution i

Solution

Write

q (x, y) = x^{2} + y^{2}

. Equality of

q

-values defines an equivalence relation induced by a function: equality is reflexive, symmetric, and transitive, so all three properties hold for

R

. The equivalence class of

(x, y)

is

[(x, y)] = {(u, v) : u^{2} + v^{2} = x^{2} + y^{2}} .

For

(x, y) \neq = (0, 0)

this is the circle centred at the origin with radius

x^{2} + y^{2}

; the remaining equivalence class is the singleton

{(0, 0)}

.

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