 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ia/paper-2/2d/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

An equivalence relation is reflexive, symmetric, and transitive. Here

∣ x - y ∣^{2} < ∣ x ∣^{2} + ∣ y ∣^{2} ⟺ Re (x \overline{y}) > 0.

The relation is reflexive and symmetric, but it is not transitive. For example, take

x = 1, y = e^{iπ /3}, z = e^{2 iπ /3} .

x \overline{y}

and

y \overline{z}

are positive, whereas that of

x \overline{z}

is negative. Hence this is not an equivalence relation.

 New to topics? Read the docs here!