Mathematics 242 – Principles of Analysis Discussion 1 – Sets and Functions September 2, 2005

Background

Today, we will review (or learn) some language about sets and functions. To begin, for us, a set will be a collection of objects, called the elements of the set. Usually, but not always, the elements of a set will be mathematical “things” like numbers, geometric figures, and so forth. Usually sets are named by capital letters like A, B, X, Y, and elements are named a, b, x, y, etc. The statement that x is an element of the set A is written in symbolic form as x ∈ A (the symbol ∈ means “is an element of”). We can specify sets by either

• listing the elements,

• giving a “recipe” for constructing all the elements, or

• giving a “test” for whether an element of a larger “universal” set is contained in the set we are talking about.

For instance, consider the set of natural numbers S = {1, 3, 5, 7, 9}. We could also define the same set as

S = {2k + 1 : 0 ≤ k ≤ 4, k ∈ N}

or as

S = {n ∈ N : n is odd and n ≤ 9}

(among many other ways). Today, we will look at a number of examples.

Discussion Questions

A) Given sets A, B inside some universal set X, we can make other sets called the union A ∪ B, the intersection A ∩ B, and the complement A^{c} (complement inside X). How are each of these defined? If you don’t know or don’t remember, look in section 1.2 of the text, and explain what is there in your own words.

B) Let the universal set be X = Z (the set of integers) and let

A = {n ∈ Z : n^{2} < 12} B = {n ∈ Z : 1 < 2n + 5 < 20}

1) List the elements in A, B.

2) What are the sets A ∩ B, A ∪ B and A^{c}?

3) Show that (A ∪ B)^{c} is the same set as A^{c} ∩ B^{c} by describing the elements of each.

C) For each natural number n, let A_{n} = {x ∈ R : x ≥ n}. (We could also write the set A_{n} using interval notation as A_{n} = [n, +∞).) Determine the following sets:

1) A_{1} ∩ A_{2}, A_{1} ∩ A_{2} ∩ A_{3}, A_{1} ∩ A_{2} ∩ A_{3} ∩ A_{4}.

2) What set is the intersection of all the A_{n}’s:

⋂_{n=1}^{∞} A_{n} = A_{1} ∩ A_{2} ∩ A_{3} ∩ ⋯ ?

Explain.

D) Let A, B be sets. A function f : A → B is a rule or mapping that to each element x ∈ A assigns a single element f(x) ∈ B. The set A is called the domain of f. The set of elements f(A) = {y ∈ B : y = f(x) for somex ∈ A} is called the range of f.

1) Let f : R → R be the function defined by f(x) = x^{2} + 1. What is the range of f?

2) Same question for f : [−1, 1] → R defined by f(x) = arctan(x). (Note the domain!)

3) How would you define the graph of a function using set notation? (What new type of object is involved?)

4) Consider the function f : R → R defined by

f(x) = \{ \begin{matrix}x & if x ∈ Q \\ 0 & if x ∉ Q \end{matrix}

Draw (or describe in words) the graph of this function.

Assignment

Writeups due in class on Monday, September 5.