1. ## Relations Question

Let R be a reflexive and transitive relation on A. Define E in A by aEb if and only if aRb and bRa. Show that E is an equivalence relation on A. Define the relation R/E in A/E by ([a]_E)R/e([b]_E) if and only if aRb. Show that the definition does not depend on the choice of representatives for [a]_E and [b]_E. Prove that R/E is an ordering of A/E.

_ is sub.

Any ideas?

2. Originally Posted by hammertime84
Let R be a reflexive and transitive relation on A. Define E in A by aEb if and only if aRb and bRa. Show that E is an equivalence relation on A. Define the relation R/E in A/E by ([a]_E)R/e([b]_E) if and only if aRb. Show that the definition does not depend on the choice of representatives for [a]_E and [b]_E. Prove that R/E is an ordering of A/E.

_ is sub.

Any ideas?
Yes! Use the definition of "equivalence relation"!

1) For all x, xEx which is true if and only if xRx and xRx. How do you know those are true?

2) For all x, y, if xEy then yEx. If yEx then xRy and yRx. Does it follow that yEx?

3) for all x, y, z, if xEy and yEz then xEz. If xEy and yEz then xRy and yRx, yRz and yRz. Does it follow that xRz and zRx?