Definition 1. If A and B are sets, then their symmetric difference is defined to be AAB (AB) U (BA) i.e. AB is the set of elements which lie in exactly one of the sets A or B. Exercise 2. Let be the relation on P(N) defined by AB AB and min(AAB) = A. (i) Prove that is a linear ordering of P(N). (ii) Prove that is not a well-ordering of P(N). Exercise 3. Prove that if m, n Ew satisfy m < n, then there exists pЄ w such that nm+p+. (Hint: Argue by induction on n.)

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.