Mathematics 242 – Principles of Analysis Discussion 1 – Set Operations September 8, 2004

Background

The language of (modern) mathematics is the language of sets. Today, we want to review some of the basic ideas about sets and the most common operations on sets. Recall that a set is essentially just a collection of objects. To specify a set, we can either list the elements, e.g. A = {1, 2, 3, 4} for the set of natural numbers less than 5, or we can give a “test” for determining which objects are in the set. The usual notation for this is called “set-builder” notation. Here is an example:

A = {x : x is a natural number and x < 5},

or

A = {x : x ∈ N and x < 5},

(N is a common name used for the set of natural numbers. Similarly, we often write Z for the set of integers, Q for the set of all rational numbers, R for the set of all real numbers.) Either of these equations is read as “A is the set of all x such that x is a natural number and x < 5.”

The proposition that an object x is an element of the set A is written symbolically as x ∈ A, and its negation is written as x ∉ A. We emphasize that x ∈ A should be a proposition – unambiguously true or false, and not both. For instance “I am an element of the set of tall people” does not qualify (at least not without a more precise definition of what the set of tall people is, or what the exact criterion for tallness is). The set with no elements (i.e. the set where x ∈ A is always false) is called the empty set, written ∅.

Some additional notation: Given sets A, B, we can form

• the union

A ∪ B = {x : x ∈ A or x ∈ B},

• the intersection

A ∩ B = {x : x ∈ A and x ∈ B},

• the set difference

A \ B = {x : x ∈ A and x ∉ B}.

Note the logical connectives used here.

For example, let A = {1, 2, 5, 6}, B = {2, 5, 7}, C = {2, 3, 4} (all subsets of the set of natural numbers). Then

\begin{aligned}A ∪ B & = \{1, 2, 5, 6, 7\} \\ B ∩ C & = \{2\} \\ A \ C & = \{1, 5, 6\} \end{aligned}

Discussion Questions

I. Using set-builder notation, how would you write: A) The set of all real numbers between 1 and 3 (inclusive)? B) The set of all natural numbers that satisfy x^{2} − 5x + 6 = 0.

II. Let A = {2, 3, 6}, B = {x : x is an even natural number and 3 < x < 7}, C = {1, 3, 6, 8}. Find A) A ∪ B. B) B \ (A ∩ C) C) (B \ A) ∩ C.

III. Using the same A, B, C from question II, decide whether the following statements are true or false, and give a reason for your conclusion in each case. Notation: X ⊆ Y is the proposition “the set X is contained in the set Y ” – that is, every element in X is also an element of Y.

A) (A ∩ B) ⊆ C B) ∅ ∈ A (What is the difference between saying ∅ ⊆ A and ∅ ∈ A?) C) (B \ A) ⊆ C D) A ∩ B ∩ C = 6. (Careful, what “type of object” is 6? Is it the same type of object as

the left side of the equation?)

IV. The set operations above satisfy a number of different properties that will be useful to us. See Theorem 5.13 in the text for a few. To practice some of the ideas about proofs we discussed for the past few days, let’s consider 5.13 g: For all sets A, B, C,

(1)  A \ (B ∩ C) = (A \ B) ∪ (A \ C)

(one of “DeMorgan’s Laws”). To prove an equality of sets X = Y, one basic approach is to show both inclusions X ⊆ Y and Y ⊆ X. This will give the basic outline of the proof, one paragraph to show each inclusion. Within each paragraph we will use another standard idea about showing inclusions: A) Explain why the statement X ⊆ Y is equivalent to the statement x ∈ X ⇒ x ∈ Y. B) “Fill in the blanks” to give a complete proof of (1):

Proof: First we show A \ (B ∩ C) ⊆ (A \ B) ∪ (A \ C). Let x ∈ A \ (B ∩ C). We want to show that x ∈ (A \ B) ∪ (A \ C) follows from this. By the definition of set difference, x ∈ A \ (B∩C) means x ∈ and x ∉ . By 1.6 a on page 6 of the text, x ∉ B∩C is equivalent to x ∉ B x ∉ C. But then x ∈ A and x ∉ B, or x ∈ A and x ∉ C. By definition, again, this means x ∈ A \B, or x ∈ A \C. Hence by the definition of the union,

. Next, we show (A \ B) ∪ (A \ C) ⊆ A \ (B ∩ C). Let . Then x ∈ A and

x ∉ B, or x ∈ A and . By the definition of the set difference, this says x ∈ A and either x ∉ B, or x ∉ C (or both). By 1.6 a again, we get x ∈ A\(B ∩ C).

C) Develop your own proof that if A, B are any sets and A ∩ B = A, then A ⊆ B, and

conversely if A ⊆ B, then A ∩ B = A. Assignment

Group write-ups due Monday, September 13.