Distinct equivalent classes the null set

**adam_leeds** · December 30th 2009, 02:33 AM

Let R be an equivalence relation on a set S. Let E and F be two distinct equivalence classes of R. Prove that E and F = null set.

Do i show that itys transitive im a bit stuck. Help much appreciated thanks.

**Swlabr** · December 30th 2009, 02:43 AM

Originally Posted by adam_leeds

Let R be an equivalence relation on a set S. Let E and F be two distinct equivalence classes of R. Prove that E and F = null set.

Do i show that itys transitive im a bit stuck. Help much appreciated thanks.

Do you mean $E \cap F = \emptyset$ ?

Assume there exists an element in $E \cap F$ , call it $x$ . Then you can apply transitivity as every element of $E$ is equivalent to $x$ , as is every element of $F$ . $e \sim x$ and $x \sim f \Rightarrow e \sim f \Rightarrow E = F$ , a contradiction.

**adam_leeds** · December 30th 2009, 02:45 AM

Originally Posted by Swlabr

Do you mean $E \cap F = \emptyset$ ?

Assume there exists an element in $E \cap F$ , call it $x$ . Then you can apply transitivity as every element of $E$ is equivalent to $x$ , as is every element of $F$ . $e \sim x$ and $x \sim f \Rightarrow e \sim f \Rightarrow E = F$ , a contradiction.

Yep thats what i meant thanks.

**adam_leeds** · January 5th 2010, 10:21 AM

Originally Posted by Swlabr

Do you mean $E \cap F = \emptyset$ ?

Assume there exists an element in $E \cap F$ , call it $x$ . Then you can apply transitivity as every element of $E$ is equivalent to $x$ , as is every element of $F$ . $e \sim x$ and $x \sim f \Rightarrow e \sim f \Rightarrow E = F$ , a contradiction.

acually im a bit confused how is this a contradiction, it doesnt show its in the null set, just that e is in f?

**Defunkt** · January 5th 2010, 10:50 AM

Originally Posted by adam_leeds

acually im a bit confused how is this a contradiction, it doesnt show its in the null set, just that e is in f?

You assumed that there is some element in $E\cap F$ . As Swlabr showed, this implies $E=F$ , however you took E,F to be distinct equivalence classes, ie. $E \neq F$ - so this is a contradiction.

**Drexel28** · January 5th 2010, 12:47 PM

Originally Posted by adam_leeds

Let R be an equivalence relation on a set S. Let E and F be two distinct equivalence classes of R. Prove that E and F = null set.

Do i show that itys transitive im a bit stuck. Help much appreciated thanks.

More of a forward-knowledge looking back approach (since you need to do this problem to prove what I'm about to say), but a relation $R$ on $S$ induces a partition $\pi$ of $S$ where the blocks are the equivalence classes. If that is a definition in your book the answer follows immediately.

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