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24 - b are in some sense indistinguishable ²or example...

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Relations with these properties occur naturally: the equality relation on sets is reflexive, symmetric and transitive; the relations on numbers and on sets are reflexive and transitive, but not symmetric; and the relation < on numbers is transitive, but not reflexive nor symmetric. Another way to define these relations is in terms of the operations on relations introduced in the previous section. P ROPOSITION 3.12 let R be a binary relation on A . 1. The relation R is reflexive if and only if id A R . 2. The relation R is symmetric if and only if R = R - 1 . 3. The relation R is transitive if and only if R R R . Proof The proof is easy and is left as an exercise. 3.6 Equivalence Relations We think of an equivalence relation as a weak equality: a R b means that a and
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Unformatted text preview: b are in some sense indistinguishable. ²or example, imagine that we have a set o± programs and we have various demands to make o± them: ±or example, we might require that the programs • always terminate; • cost less than a hundred pounds; • compute π to 100 decimal places; • . . . Even though two programs are not equal, they can satis±y the same demands and so be ‘equal enough’ ±or our purposes. In such a case, we say that two programs are equivalent . D EFINITION 3.13 Let A be a set and R a binary relation on A . The relation R is an equivalence relation i± and only i± R is refexive, symmetric and transitive. We sometimes just say that R is an equivalence . 25...
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