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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Relations with these properties occur naturally: the equality relation on sets is refexive, symmetric and transitive; the relations on numbers and on sets are refexive and transitive, but not symmetric; and the relation < on numbers is transitive, but not refexive nor symmetric. Another way to deFne these relations is in terms o± 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 refexive i± and only i± id A R . 2. The relation R is symmetric i± and only i± R = R - 1 . 3. The relation R is transitive i± and only i± R R R . Proof The proo± is easy and is le±t as an exercise. 3.6 Equivalence Relations We think o± an equivalence relation as a weak equality: aRb means that a and
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: b are in some sense indistinguishable. or 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 to 100 decimal places; . . . Even though two programs are not equal, they can satisy 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...
View Full Document

This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

Ask a homework question - tutors are online