Understanding Sets: Notation, Unions, and Intersections

Study Notes

A set is a collection of objects, with the objects being called elements of the set.
Set notation forms the basis of mathematical communication, being concise and precise.
"α ε B" means "a is an element of B".
"BC" means "B is a subset of C".
"C2B" means "C contains B as a subset".
Putting a slash through a symbol negates it; for example, "q∉B" means "q is not an element of B".
A set of A = {1,2,3} and a set of B ={1,2,a,b }, and a set of c ={ a, b} are examples of sets.

Unions of Sets:

A U B means {things that are in A or B}, which can be written as { x | x ∈ A or x ∈ B }.
This is the set of x such that x is an element of A, or x is an element of B.

Intersection of Sets:

A ∩ B includes things that are in both A and B, and it can be represented as { x | x ∈ A and x ∈ B }.
This is the set of X such that is an element of A and X is also an element of B.
For example, given A = {1,2,3}, B = {1,2,a,b}, and C = {a,b}, A U B = {1,2,3,a,b}, A ∩ B = {1,2}, and B ∩ C = {a, b}.
Ø represents the empty set.

Set Complement:

A \ B includes things that are in A but not in B, represented as { x | x ∈ A and x ∉ B }.
This is the set of x such that x is an element of A, and x is not an element of B.
For example, given A = {1,2,3} and B = {1,2,a,b}, A\B = {3} and B\A = {a,b}.

Sets of Numbers:

The set of natural numbers is symbolized by N = {0, 1, 2, 3,...}.
The set of integers is Z = {..., -2, -1, 0, 1, 2, 3,...}.
The set of rational numbers is Q = { a/b | a ∈ Z and b ∈ Z, b ≠ 0 }.
The set of real numbers is denoted by R.
Real numbers (R) are ordered: if a, b ∈ R, then either a < b, a > b, or a = b.
Each real number has a modulus, called absolute value.
This is the distance from a to 0 on the number line.

Intervals (Subsets of R):

[a, b] = {x ∈ R | a ≤ x and x ≤ b}
(a, b) = {x ∈ R | a < x and x < b}
[a, b) = {x ∈ R | a ≤ x and x < b}
[a, ∞) = {x ∈ R | x ≥ a}
(a, ∞) = {x ∈ R | x > a}
(-∞, b] = {x ∈ R | x ≤ b}
∞ and -∞ are not elements of R, they are convenient notations.

Description

Learn about set notation, including elements, subsets, unions, and intersections. This lesson provides definitions and examples to clarify set theory concepts. Grasp the fundamentals of mathematical communication using sets.