Properties of relations

Give examples of relations with the following properties:

1. Reflexive but not symmetric

2. Symmetric but not transitive

3. Transitive, symmetric, but not reflexive

Show that a relation can be both symmetric and anti-symmetric by giving an interpretation that makes both of the following sentences true:

$\displaystyle\forall x\forall y(R(x,y)\rightarrow R(y,x))$ $\displaystyle\forall x\forall y((R(x,y)\wedge R(y,x))\rightarrow x=y)$