Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment I.

Notation. Let X be a set. By P (X) we will denote the collection of all subsets of X.

1. Problem 1. Let X consist of n elements. Prove that P (X) consists of 2^{n} elements. Definition 1. A nonempty subcollection R ⊂ P (X) is called a ring of subsets of X, or just a ring, provided R is closed under the union, intersection, and difference. In other words, for any A, B ⊂ R, A ∩ B ⊂ R, A ∪ B ⊂ R, A \ B ⊂ R. Observe that this implies also A△B := (A \ B) ∪ (B \ A) ⊂ R (the last operation is called the symmetric difference). Examples. For any X, P (X) is a ring. A set of all bounded subsets of the real line is a ring. A collection {A, ∅}, (where ∅ is an empty set), is a ring.

2. Problem 2. Let a subcollection R ⊂ P (X) be such that for any A, B ⊂ R, A∩B ⊂ R, and A ∪ B ⊂ R. Prove that R is not (in general) a ring. On the other hand, if for any A, B ⊂ R, A ∪ B ⊂ R, and A \ B ⊂ R, then R is a ring. Definition 2. A nonempty subcollection ℵ ⊂ P (X) closed under intersection is called a subring , if ∅ ⊂ ℵ, and if A, A_{1} ⊂ ℵ, A_{1} ⊂ A, then A = ∪_{k=1}^{n}A_{k}, where all A_{k} are disjoint.

3. Problem 3. Prove that the set of all segments (open, closed, semi-open) on the real line is a subring, but not a ring.

4. Problem 4. Let X := {a, b, c}, be a set consisting of three elements.

a) Write out P (X).

b) Give an example of a subring, consisting of elements of P (X), which is not a ring.

c) Discribe all subrings, which can be obtained from the elements of X.

d) Discribe all rings, which can be obtained from the elements of X. Definition 3. Let A ⊂ X. A function

1_{A}(x) := \{ \begin{matrix}1 & for x ∈ A \\ 0 & for x ∉ A, \end{matrix}

defined on X, is a called a characteristic function of a set A.

5. Problem 5. Prove that for A, B ⊂ X, a) 1_{A∩B}(x) = 1_{A}(x)1_{B}(x),

b) 1_{A∪B}(x) = 1_{A}(x) + 1_{B}(x) − 1_{A}(x)1_{B}(x),

c) 1_{A△B}(x) = 1_{A}(x) + 1_{B}(x),

d) 1_{A\B}(x) = 1_{A}(x) − 1_{A}(x)1_{B}(x).

6. Problem 6. Let R be a collection of subsets of X, and let R̃ be a collection of characteristic functions of sets belonging to R. Prove that R is a ring of subsets of X if and only if R̃ is a usual ring of functions (in other words, if 1_{A}(x), 1_{B}(x) ∈ R̃, then 1_{A}(x)1_{B}(x) ∈ R̃, 1_{A}(x) + 1_{B}(x) ∈ R̃).

7. Problem 7*. Assume that you have 10 sets. How many disjoint nonempty sets you can obtain by taking union, intersection, difference, and symmetric difference of the given sets? You may apply these operations as many times as you want.

Hint: Consider the sets A_{1}, A_{2}, ..., A_{10} of sequences of length 10 consisting only of 1, 0, where all sequences belonging to A_{i} have 1 in the i−th place, i = 1, ..., 10.