$\newcommand{\defeq}{\mathrel{\mathop:}=}$

## 2008/09/15

### 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

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!

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