You are currently browsing the daily archive for July 9, 2009.

High-school students and undergraduates are (almost) always taught the following definition of an equivalence relation.

A binary relation R on a set A is an equivalence iff it satisfies

  • the reflexive property: for all a  in A, a R a,
  • the symmetric property: for all a, b in A, if a R b, then b R a, and
  • the transitive property: for all a, b, c in A, if a R b and b R c, then a R c.

However, there is another formulation of an equivalence relation that one usually doesn’t hear about, as far as I know. And, it is the following one.

A binary relation R on a set A is an equivalence iff it satisfies

  • the reflexive property: for all a  in A, a R a, and
  • the euclidean property: for all a, b, c in A, if a R b and a R c, then b R c.

Exercise:  Show that a binary relation R on a set A is reflexive, symmetric and transitive iff it is reflexive and euclidean.


Our other blog

Visitors to this blog

Blog Stats

  • 355,659 hits

Wikio Ranking

Wikio - Top Blogs - Sciences

Current Online Readers

Recent Comments

John Favors on Solution to POW-13: Highly…
Wayne J. Mann on Solution to POW-12: A graph co…
erneststephen on The 54th Carnival of Math…
anhtraisg on p^q + q^p is prime
prof dr drd horia or… on My first post
prof drd horia orasa… on My first post
prof dr mircea orasa… on Inequality with log
notedscholar on Self-referential Paradoxes, In…
prof dr mircea orasa… on Inequality with log
prof dr mircea orasa… on Inequality with log
prof dr mircea orasa… on 2010 in review
kenji on Basic category theory, I
prof dr mircea orasa… on Solution to POW-10: Another ha…
prof drd horia orasa… on Continued fraction for e
prof drd horia orasa… on Inequality with log

Archives

July 2009
M T W T F S S
 12345
6789101112
13141516171819
20212223242526
2728293031