# MATH 360 Review of Set Theory

**Definition:** A **set** is a collection of objects. The
objects that make up a set are

called its **elements**. Capital letters are used for sets, and lowercase for
elements. If

A is a set, then

• x ∈ A means x is an element of A.

• x A means x is not an element of A.

**Example: **Let A = {−1, 0, 2,Rebeccala}. Then −1 ∈ A, 3 A, Rebeccala ∈ A.

Recall the following important sets:

• Z = the integers = {0,±1,±2, ...}

• = the positive integers ={1, 2, 3, 4, ...}

• N = the natural numbers =

• Q = the rationals = { : a, b ∈ Z, b
≠ 0}

• P = the irrationals = numbers that cannot be expressed as a rational

• R = the real numbers = {x : x ∈ Q or x ∈ P}

• R+ = the positive real numbers = {x ∈ R : x > 0}

• Also, recall interval notation:

– (a, b) = {x ∈ R : a < x < b}

– [a, b] = {x ∈ R : a ≤ x ≤ b}

– (a,∞) = {x ∈ R : x > a} etc. Note that
= (0,∞)

Interval notation should be quite familiar to you from your Calculus courses,

but if it is **not**, please come to see me - we'll go over it!

• The empty set, , is the set containing no elements. Sometimes is written as

{ }. Note: {} ≠ !

**Definition:** Let A and B be sets. Then A is a **subset **of B, written A B, if
every

element of A is also an element of B. Formally, this means that if x ∈ A, then x
∈ B.

Thus, to have A B means that there is at least one element of A that is not
also

in B, i.e. there exists x ∈ A such that x B.

**Example: **A = {1, 2, 3, 4},B = {2, 4},C = {1, 3, 4}. Then B
A and C A, but

**Definition:** Two sets A and B are **equal,** written A = B, provided that A B and

B A.

**Definition:** The **union** of two sets A and B is A ∪ B = {x : x ∈ A or x ∈ B}. The

**intersection** of two sets A and B is A ∩ B = {x : x ∈ A and x ∈ B}.

**Example:** Let A = (0, 3],B = [3,∞),C = {0, 6,Rebeccala}. Then A ∪ B =

(0,∞),A ∩ B = {3},A ∩ C = .

**Definition:** The sets A and B are** disjoint** if A ∩ B = . Thus the sets A and C of

the example above are disjoint.

**Definition:** The ** set difference ** of A and B, A−B = {x : x ∈ A and x B}. Note:

B does not have to be a subset of A in order to form A − B. This is sometimes
also

written as A\B.

Example: Let A = [0, 4],B = (1, 5],C = (3, 6),D = (1,∞). Then B − A =

(4, 5],A − B = [0, 1],C − D = ,D − C = (1, 3] ∪ [6,∞).

**Definition: **The ** product ** (or Cartesian product) of sets A and B is the set A×B =

{(x, y) : x ∈ A and y ∈ B}. Note: the notation (x, y) here means the ordered
pair -

not the open interval. (Tricky! )

**Examples:**

1. If A = [1, 4],B = [2, 3], then A × B =

B × A =

2. If A = {1, 2},B = {3, 4, 5}, then A×B = {(1, 3), (1,
4), (1, 5), (2, 3), (2, 4), (2, 5)}.

Thus if A and B are finite sets with m and n elements, respectively, then A×B

has mn elements. (This can be proven by induction.)

3. If A = [1, 3),B = [1, 2), then A × B =

4. If A = (1,∞),B = (1, 2), then A × B =

When drawing by hand , I use solid lines for the ”[” and
dotted lines for the ”(”

and arrows to indicate when it is continuing to infinity.

Prev | Next |