Sets and Set Theory Flashcards
Sets
Unordered collections of objects that can be finite or infinite.
Elements
Members of the set
Singleton set
Set with exactly one element.
The ways to specify sets
- Description method
- Roster/Enumeration method
- Rule method (Set Builder Notation)
V = set of all vowels in the English alphabet
Description method.
V = {a, e, i, o, u}
Roster/Enumeration method.
A = {x|x is a perfect square less than 10}
Rule method.
Repeated elements are to be written _.
once
Cardinality
Number of elements in a set.
Symbol for cardinality
||
|S| could be _ or _, depending on the number of elements
finite, infinite
|{}|
0
|{{}}|
1
ℕ
Set of natural numbers = {1, 2, 3, …}
W
Set of whole numbers = {0, 1, 2, …}
ℤ
{ …, -2, -1, 0, 1, 2, … } = set of integers
ℤ+
{ 1, 2, 3, … } = set of positive integers
ℤ-
{ …, -3, -2, -1 } = set of negative integers
ℚ
{ p/q | p,q E ℤ and q≠0 } = set of rational numbers
ℝ
Set of real numbers
ℂ
{ a + bi | a,b E ℝ, i = √(-1) } = set of complex numbers
A ⊆ B
Every element of A is an element of B
B ⊇ A
If and only if A ⊆ B
A = B
Every element of A is an element of B and every element of B is an element of A
A ⊂ B
Every element of A is an element of B and B has at least one element that is not in A
B ⊃ A
If and only if B ⊇ A and there exists an element that is in set B but not in A
⊇
Superset
⊃
Proper superset
⊆
Subset
⊂
Proper subset
Venn Diagrams
Used to visualise relationships of sets
Disjoint sets
Sets that have no common element
A U B
The elements are those in A or in B
A ∩ B
The elements are those in A and in B
A ∩ B are disjoint if…
A ∩ B = {}
A - B
Set Difference. The elements are those in A and not in B
A’
The elements are in U and not in A
Other ways to write A’
-A, or U-Ā
{}’
U
A U {}
A
{} ∩ U
{}
A - A
{}
Identities for Set Difference: If set A and B are equal then…
A–B = A–A = {}
Identities for Set Difference: When an empty set is subtracted from a set…
the result is that set itself (A – {} = A)
Identities for Set Difference: When a set is subtracted from an empty set…
the result is an empty set ( {} – A = {} }
Identities for Set Difference: When a superset is subtracted from a subset, the result is…
empty set (A – B = {}, if A ⊂ B)
Identities for Set Difference: If A and B are disjoint sets then…
A - B = A, and B - A = B
Logic and Sets Correspondence: v = _
U
Logic and Sets Correspondence: ^ = _
∩
Logic and Sets Correspondence: ~ = _
—
Logic and Sets Correspondence: T = _
U
Logic and Sets Correspondence: F = _
{}
Logic and Sets Correspondence: Truth Table = __
Membership Table
Logic and Sets Correspondence: T = _ or _
E, 1
Logic and Sets Correspondence: F = _ or _
∉, 0
(x E A) is the same as…
~ (x E A), or x E A’
Irrational numbers
Any real number that cannot be expressed as the quotient of two integers.
Composite numbers
A positive integer that has at least one divisor other than 1 and itself. A positive integer that can be formed by multiplying two smaller positive integers.
Prime numbers
Numbers that have only 2 factors: 1 and themselves.
