Download Page 1 of 4 Math 3336 Section 2.1 Sets • Definition of sets

Math 3336
Section 2.1
Sets
•
•
•
•
•
•
Definition of sets
Describing Sets
• Roster Method
• Set-Builder Notation
Some Important Sets in Mathematics
Empty Set and Universal Set
Subsets and Set Equality
Cardinality of Sets
Definition: A set is an unordered collection of distinct objects.
Examples:
a. the students in this class
b. the chairs in this room
c. counting numbers
Definition: The objects in a set are called the elements, or members of the set. A set is said to
contain its elements.
Notation:
• The notation 𝑎𝑎 ∈ 𝐴𝐴 denotes that 𝑎𝑎 is an element of the set 𝐴𝐴.
• If 𝑎𝑎 is not a member of 𝐴𝐴, write 𝑎𝑎 ∉ 𝐴𝐴.
Ways to describe a set
1. Roster method
• List all elements of a set
• Order NOT important
• Each distinct object is either a member or not; listing more than once does not
change the set.
• Ellipses (…) may be used to describe a set without listing all of the members
when the pattern is clear.
Example: 𝑆𝑆 = {𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑}
Usually elements in a set have a common property but this is NOT necessary.
Example: 𝐶𝐶 = {I, love, math, 10, Houston}
2. Set builder notation
• Specify a property that all elements in a certain set must have.
Example: 𝐵𝐵 = {𝑥𝑥 |𝑥𝑥 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 12
Page 1 of 4
Some important sets
ℕ = natural numbers = {0,1,2,3 … . }
ℝ = set of real numbers
ℝ+ = set of positive real numbers
ℤ = integers = {… , −3, −2, −1,0,1,2,3, … }
ℤ+ = positive integers = {1,2,3, … . . }
ℂ = set of complex numbers.
ℚ = set of rational numbers
Example: 𝐵𝐵 = {𝑥𝑥 |𝑥𝑥 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 12}
Example: {ℕ, ℤ, ℚ, ℝ}
Definition: Two sets are equal if and only if they have the same elements.
Notation: 𝐴𝐴 = 𝐵𝐵 𝑖𝑖𝑖𝑖𝑖𝑖 ∀𝑥𝑥 (𝑥𝑥 ∈ 𝐴𝐴 ↔ 𝑥𝑥 ∈ 𝐵𝐵)
Example: 𝐴𝐴 = {𝑟𝑟𝑟𝑟𝑟𝑟, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏}, 𝐵𝐵 = { 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏}
Example: 𝐴𝐴 = {𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟}, 𝐵𝐵 = {𝑟𝑟𝑟𝑟𝑟𝑟}
Venn Diagrams
Sets can be represented graphically using Venn diagrams. Venn diagrams are often used to show
relations between sets.
John Venn (1834-1923)
Cambridge, UK
The universal set 𝑈𝑈 is the set containing everything currently under consideration.
 Sometimes implicit
 Sometimes explicitly stated.
 Contents depend on the context.
The empty (null) set is the set with no elements. Symbolized ∅, but {} also used.
Page 2 of 4
Example: Draw a Venn diagram that represents 𝑉𝑉 = {𝑎𝑎, 𝑒𝑒, 𝑖𝑖, 𝑜𝑜, 𝑢𝑢}.
Example: 𝐴𝐴 = {𝑥𝑥 ∈ ℕ |𝑥𝑥 < 0}
Definition: A set with one element is called a singleton set.
Example: 𝐴𝐴 = {𝑥𝑥 ∈ ℕ|𝑥𝑥 ≤ 0}
Definition: The set 𝐴𝐴 is a subset of 𝐵𝐵 if and only if every element of 𝐴𝐴 is also an element of 𝐵𝐵.
𝐴𝐴 is a strict (proper) subset of 𝐵𝐵 if 𝐴𝐴 is a subset of 𝐵𝐵 and 𝐴𝐴 is not equal to 𝐵𝐵.
Notation: 𝐴𝐴 ⊆ 𝐵𝐵 and 𝐴𝐴 ⊂ 𝐵𝐵 respectively.
Example: {1, 2} ⊂ {1, 2, 3, 4} ⊆ {1, 2, 3, 4}
•
•
•
To show that 𝐴𝐴 ⊆ 𝐵𝐵, show that if 𝑥𝑥 belongs to 𝐴𝐴, then x also belongs to 𝐵𝐵.
To show that 𝐴𝐴 ⊈ 𝐵𝐵, find a single 𝑥𝑥 ∈ 𝐴𝐴 such that 𝑥𝑥 ∉ 𝐵𝐵.
For every set 𝑆𝑆, ∅ ⊂ 𝑆𝑆 and 𝑆𝑆 ⊆ 𝑆𝑆.
Prove: For every set 𝑆𝑆, ∅ ⊂ 𝑆𝑆.
Proof:
Definition: Given a set A, cardinality (size) of 𝐴𝐴 is the number of elements in 𝐴𝐴.
Notation: |𝐴𝐴|
Definition: The power set is the set of all subsets of a given set, and is denoted by 𝒫𝒫(𝐴𝐴).
If a set has 𝑛𝑛 elements its power set contains 2𝑛𝑛 elements.
Page 3 of 4
Try this:
𝐴𝐴 = {1, {2}, {3,4}}, 𝐵𝐵 = {1, 2, 3, 4}
1. True or False?
a. 1 ∈ 𝐴𝐴
b. 2 ∈ 𝐴𝐴
c. {3} ⊂ 𝐴𝐴
d. 2 ∈ 𝐵𝐵
e. {3} ⊂ 𝐵𝐵
f. ∅ ∈ 𝐴𝐴
g. ∅ ⊂ 𝐵𝐵
h. 𝐴𝐴 ⊆ 𝐵𝐵
2. Find
a.
b.
c.
d.
e.
f.
g.
h.
|𝐴𝐴|
|𝐵𝐵|
𝒫𝒫(𝐴𝐴)
𝒫𝒫(𝐵𝐵)
|∅|
|{∅}|
𝒫𝒫(∅)
𝒫𝒫({∅})
Page 4 of 4