19-equivalence-relations

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

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1 Equivalence Relations Aaron Bloomfield CS 202 Epp, section ???
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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
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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
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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 ) means that m | a-c
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