Lecture Notes 7 - Equivalence Relations...

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Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide01.html 1 of 1 09/11/2003 03:48 PM prev | slides | next Equivalence Relations 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide02.html 1 of 1 09/11/2003 03:48 PM Equivalence Relations prev | slides | next Consider the following relations on the set of people in this room {( a , b ) | a and b were born in the same month}, {( a , b ) | a and b are the same sex}, {( a , b ) | a and b are from the the same state}. Observe that these relations are all reflexive, symmetric and transitive. Because of this they are all equivalent in some way. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide03.html 1 of 1 09/11/2003 03:48 PM Equivalence Relations prev | slides | next A relation on a set A is an equivalence relation if it is reflexive, symmetric and transitive. Suppose that R is a relation on the positive integers such that ( a , b ) R if and only if a <5 and b <5. Is R and equivalence relation? Since a = a it follows that if a <5 then ( a , a ) R so we know that R is reflexive. Suppose ( a , b ) R so both a <5 and b <5. In this case certainly ( b , a ) R so that R is symmetric. Finally, if ( a , b ) R and ( b , c ) R then both a and c are less than 5 so ( a , c ) R showing that R is transitive. Thus R is an equivalence relation. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide04.html 1 of 1 09/11/2003 03:48 PM Equivalence Relations prev | slides | next Let R be an equivalence relation on a set A . The set of all elements that are related to an element a of A is called the equivalence class of a . This is denoted [ a ] R or just [ a ] if it is clear what R is.
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