P is a preorder for a set A if P is a reflexive and transitive relation on A?

Question

Define a relation E on A by xEy iff xPy and yPx. Show that E is an equivalence relation on A.

Answer 1

You must show that E is reflexive, transitive and symmetric.

xPx because P is reflexive, so xEx.

Suppose xEy. Then xPy and yPx, but that also implies yEx, so E is symmetric.

Suppose xEy and yEz. Then xPy, yPx, yPz, and zPy. Because P is transitive, you have xPz and zPx, so xEz, therefore E is transitive.