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!
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.
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.
