19-equivalence-relations - Equivalence Relations Aaron...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Equivalence Relations Aaron Bloomfield CS 202 Epp, section ???
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Introduction Certain combinations of relation properties are very useful We won’t have a chance to see many applications in this course In this set we will study equivalence relations A relation that is reflexive, symmetric and transitive Next slide set we will study partial orderings A relation that is reflexive, antisymmetric, and transitive The difference is whether the relation is symmetric or antisymmetric
Background image of page 2
3 Equivalence relations A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive Consider relation R = { ( a,b ) | len ( a ) = len ( b ) } Where len ( a ) means the length of string a It is reflexive: len ( a ) = len ( a ) It is symmetric: if len ( a ) = len (b), then len ( b ) = len ( a ) It is transitive: if len ( a ) = len (b) and len (b) = len (c), then len (a) = len (c) Thus, R is a equivalence relation
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 Equivalence relation example Consider the relation R = { ( a,b ) | m | a-b } Called “congruence modulo m Is it reflexive: (a,a) R means that m | a-a a-a = 0, which is divisible by m Is it symmetric: if ( a,b ) R then ( b,a ) R ( a,b ) means that m | a-b Or that km = a-b . Negating that, we get b-a = - km Thus, m | b-a , so (b,a) R Is it transitive: if ( a,b ) R and ( b,c ) R then ( a,c ) R ( a,b ) means that m | a-b , or that km = a-b ( b,c ) means that m | b-c , or that lm = b-c ( a,c
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/11/2011 for the course ECON 1110 at Cornell University (Engineering School).

Page1 / 12

19-equivalence-relations - Equivalence Relations Aaron...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online