Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 4 / 2D / ii

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- Solution ii

Solution

The union need not be an equivalence relation, because transitivity can mix the two relations. Take

X = {1, 2, 3}

. Let

R

have equivalence classes

{1, 2}

and

{3}

, and

S

have classes

{1}

and

{2, 3}

. Then

1 W 2

and

2 W 3

, but neither component relates

1

to

3

, so

1 W 3

fails. The union of equivalence relations is reflexive and symmetric; it is precisely transitivity that may fail.

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