Relations and sets

1) Find a relation R on a set S that is neither Symmetric nor antisymmetric

2) Let S be a set containing exactly n elements. How many antisymmetric relations on S are there.

3) give a recursive definition of X^n for any positive integer n

4) give a recursive definition of the nth odd positive integer

5) Let g: Z -> Z be defined by g(x)= ax + b, where Z denotes the set of integers and a,b E Z with a not equal 0
    a) prove that g is one-to-one
    b) what must be true about a and b if g is onto?

6) let S be a set of people. For x, y E S, define xRy to mean that x = y or x is a decendant of y. Prove that R is a partial order on S