{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 26 - b R v Now a R x x R b and b R v so a R v by...

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

The equivalence classes of a set A can be represented by a Venn diagram: for example, A 1 A 2 A 3 A 4 A 5 A In this case, there are five equivalence classes, illustrated by the five disjoint subsets. In fact, the equivalence classes always separate the elements into disjoint subsets that cover the whole of the set, as the following proposition states formally. P ROPOSITION 3.15 The set of equivalence classes { [ a ] : a A } forms a partition of A : that is, each [ a ] is non-empty; the classes cover A : that is, A = a A [ a ] ; the classes are disjoint (or equal): a, b A. [ a ] [ b ] = ∅ ⇒ [ a ] = [ b ] . Proof Given any a A , then a R a by reflexivity and so a [ a ] . Also a a A [ a ] , and hence the classes cover A . Suppose [ a ] [ b ] = , and let x [ a ] [ b ] . This means that a R x and b R x . It follows that x R b by symmetry. Given any v [ b ] , observe that
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: b R v . Now a R x , x R b and b R v , so a R v by transitivity. Therefore v ∈ [ a ] and so [ b ] ⊆ [ a ] . But [ a ] ⊆ [ b ] using a similar argument, so [ a ] = [ b ] . ± 3.7 Transitive Closure Consider the following situation. There are various ±ights between various cities. ²or any two cities, we wish to know whether it is possible to ±y from one to the other allowing for changes of plane. We can model this by deFning a set City of cities and a binary relation R such that a R b if and only if there is a direct ±ight from a to b . This relation may be represented as a directed graph with the cities as nodes, as in the following example: 27...
View Full 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