# Equivalence Relation Proof

• 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.

• 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 \$\displaystyle x\sim y\$. The from being reflexive we know that \$\displaystyle x\sim x\$

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

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

From that, how and why do we know that \$\displaystyle 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.