Paper 4, Section I, 2D2 \mathrm{D}

Numbers and Sets | Part IA, 2012

What is an equivalence relation on a set XX ? If RR is an equivalence relation on XX, what is an equivalence class of RR ? Prove that the equivalence classes of RR form a partition of XX.

Let RR and SS be equivalence relations on a set XX. Which of the following are always equivalence relations? Give proofs or counterexamples as appropriate.

(i) The relation VV on XX given by xVyx V y if both xRyx R y and xSyx S y.

(ii) The relation WW on XX given by xWyx W y if xRyx R y or xSyx S y.

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