Reference no: EM131015100

Part A-

1) Let f = {x ∈ R |0 < x}. For each i ∈ I, let

D_i = {(x, y) ∈ R x R|0 < x < I and 0 < y < 1/i}.

Find the following. If necessary, you may describe your answers geometrically.

a) ∪_i∈I D­i

b) ∩_i∈I D_i

2) Let I be an indexed set and A_i be a collection of sets with i ∈ I. Prove DeMorgan's Laws for indexed sets, that is:

a) ∪_i∈I A_i^c = (∩_i∈IA_i)^c

b) ∩_i∈I A_i^c = (∪_i∈I A_i)^c

Part B-

1 (Sundstrom) Let S be the function that associates with each natural number the set of its natural number divisors. For example, S(6) = {1, 2, 3, 6} and S(10) = {1, 2, 5, 10}.

a) What is the domain of S? Determine an appropriate codomain of S.

b) Determine S(n) for three prime and four composite values of n.

c) Does there exist a natural number n such that S(n) has only one element? Explain your reasoning.

d) Does there exist a natural number n such that S(n) has exactly two elements? Explain your reasoning.

e) Is the following statement true or false? Explain your reasoning.

For all m, n ∈ N, if m ≠ n, then S(m) ≠ S(n).

f) Is the following statement true or false? Explain your reasoning.

For any T ⊆ N, there exists an n ∈ N such that S(n) = T.

2) (Multivariable Functions) Let f: N x Z → Z x Z be given by f (x, y) = (x + y, xy).

a) What is f(2, 2)? What is f (1, 2)? 1(2, -1)?

b) What is f^-1[{(z, 1)|z ∈ Z}]? Why?

c) Suppose (z, 1) is in the range of f. What can you conclude about z?

d) What is f^-1 [{(z, 0)|z ∈ Z}]Why?

3) Let f : S → T be a function. Suppose A and B be subsets of S, and C and D be subsets of T.

a) Prove that C ⊆ D ⇒ f^-1[C] ⊆ f^-1[D]. Provide a counterexample to show that the converse is false.

b) Prove that f[A] - f[B] ⊆ f [A - B]. Provide a counterexample to show that f[A - B] ? f[A] - f[B].

c) Prove that A ⊆ f^-1[f[A]].  Provide a counterexample to show that f^-1 [f[A]] ? A. Notice that this means, in general, f^-1[f[A]] ≠ A.

d) Suppose S = T and that S is the universal set. Show that f[A^c] ≠f[A]^c by showing that neither set is a subset of the other. (Both can be shown with a single counterexample, although this is not required.)