Lecture Notes 7

Lecture Notes 7 - Equivalence Relations...

This preview shows pages 1–2. Sign up to view the full content.

Equivalence Relations 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 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 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 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.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern