Equivalence Relation Proof

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May 30th 2013, 10:25 AM
dpmagee10

Equivalence Relation Proof

So, here is the question:

"Suppose ~ is a relation on A that is reflexive and has the property that for all elements a, b, and c in A, if a~b and a~c, then b~c. Prove that ~ is an equivalence relation on A."

This is from Keef and Guichard's "An Introduction to Higher Mathematics," through which I am trying to slog right now (see link for PDF: http://people.whitman.edu/~gordon/higher_math.pdf).

On the text's p. 27 in ch. 1.7, #5 (written out above) is currently stumping the hell out of me. I know that Reflexivity is obviously given, but I don't know how to pull out Symmetry and Transitivity from the rest of the problem.

Can someone please help me? Thanks!
May 30th 2013, 10:39 AM
Plato

Re: Equivalence Relation Proof

Quote:

Originally Posted by dpmagee10

"Suppose ~ is a relation on A that is reflexive and has the property that for all elements a, b, and c in A, if a~b and a~c, then b~c. Prove that ~ is an equivalence relation on A."
Symmetry and Transitivity from the rest of the problem.

For Symmetry
Suppose that $x\sim y$ . The from being reflexive we know that $x\sim x$

Now we have $x\sim y~\&~ x\sim x$ . Apply the definition of the relation,

For Transitivity.
Suppose that $s\sim t~\&~t\sim u$ .

From that, how and why do we know that $t\sim s~?~$ Again apply the definition of the relation.
May 30th 2013, 10:48 AM
emakarov

Re: Equivalence Relation Proof

Quote:

Originally Posted by dpmagee10

"Suppose ~ is a relation on A that is reflexive and has the property that for all elements a, b, and c in A, if a~b and a~c, then b~c. Prove that ~ is an equivalence relation on A."

Relations satisfying the property above are sometimes called Euclidean.