1-Prove that the set operation "symmetric difference" is commutative, i.e., for any sets A and B,

A B = B A.

(The operation is defined in section 3.4 in the textbook. Do not assume that any other set operations such as "union" or "intersection" are commutative - give a complete proof with conjunctions/disjunctions of statements. Venn diagrams are not acceptable as proofs.)

2- Let x and y be real numbers. Prove or give a counterexample:

(a) x + y = x + y

(b) x x 1

(c) x = x

Hint: For many problems involving floor/ceiling functions it is useful to use a real number x as x = n + where n is an integer and 0 < 1, so that is the "decimal part" of x.

3- Let f : A B and g : B C be bijective functions. Prove that

(g f)^-1 = f^-1 g^-1

Hint: To prove that two functions are equal, pick an arbitrary point in the domain and show that both functions will give the same value. Start by figuring out what the domain is. An arrow diagram may be useful to help visualize.

4- Prove by induction that for any integer n 2, we have

1 + 1/4 + 1/9 + ... + 1/n² < 2 - 1/n^.