Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 2D / i

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- Solution i

Solution

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} .

The real parts of

x \overline{y}

and

y \overline{z}

are positive, whereas that of

x \overline{z}

is negative. Hence this is not an equivalence relation.

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