sets, relations, and equivalence classes?

Problem 1 Let S be a set with n elements and let a and b be distinct elements of S. How many relations R are there on S such that 1. (a, b) R? 3. no ordered pair in R has a as its first element? 4. at least one ordered pair in R has a as its first element? 5. no ordered pair in R has a as its first element or b as its second element? 6. at least one ordered pair in R either has a as its first element or has b as its second element? Problem 2 (Projective Plane) Denne A-R2-{(0,0)) the set of all points in the plane minus the origin. Define a relation between two points (x, y) and (r',y) by stating that they are related if and only if they lie on the same line passing through the origin 1. Show that this relation is an equivalence relation 2. draw a graphical representation of the equivalence classes by picking a representative from each class The space of all equivalence classes under this relation is called the projective plane