Simple Exercise

Prove that fº . R . f is an equivalence relation if R is. (f is, by convention, a total function.)

Proof. (i) Reflexivity:

id  ⊆  fº . f  ⊆  fº . id . f  ⊆  fº . R . f

(ii) Transitivity:

fº . R . f . fº . R . f  ⊆  fº . R . R . f  ⊆  fº . R . f

(iii) Symmetry:

(fº . R . f)º  =  fº . Rº . f  =  fº . R . f

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Example: take f = signum and R the usual equality on real numbers.

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I certainly like the way AoP treats relations. So concise!