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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2020

Paper 2

An <equivalence relation> is reflexive, symmetric, and transitive. Here
$$
|x-y|^2<|x|^2+|y|^2
\iff \operatorname{Re}(x\overline y)>0.
$$
The relation is reflexive and symmetric, but it is not transitive. For example, take
$$
x=1,\qquad y=e^{i\pi/3},\qquad z=e^{2i\pi/3}.
$$
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.

Solution

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

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

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

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