Download Sets and Set Structures

Basic Set Notation
Sets and Set Structures
An organization is a set of relationships that are persistent over
time. (Kevin Kelly)
x ∈ A means value x is an element of set A.
x 6∈ A means value x isn’t an element of set A.
There are two notations for describing sets.
Roster. A = {1, 3, 5, 7, 9}. B = {2, 3, 5, 7}.
Set Builder. C = {x | x > 0 ∧ x < 10}
Equality. A = B ≡ ∀x (x ∈ A ↔ x ∈ B)
This implies that {1, 2, 3} = {1, 2, 2, 2, 3, 3}.
Subset. A ⊆ B ≡ ∀x(x ∈ A → x ∈ B)
Ellipses. {0, 1, . . . , 9} contains 0, 1, and so on up to 9.
CS 2233 Discrete Mathematical Structures
Basic Set Notation. .
Special Sets . . . . . .
Size of Sets. . . . . . .
Sets of Sets . . . . . .
Set Operations . . . .
Set Identities. . . . . .
Venn Diagrams . . . .
Special Cardinalities .
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Sets and Set Structures – 2
Special Sets
∅ = { }. The empty set. ∀x(x 6∈ ∅)
U . The universal set. ∀x(x ∈ U )
Z. The set of integers.
N. Natural numbers. N = {x | x ∈ Z ∧ x ≥ 0}
Z+ . Positive integers. Z+ = {x | x ∈ Z ∧ x > 0}
R. The set of real numbers.
R+ . Positive reals. R+ = {x | x ∈ R ∧ x > 0}
Q. Rational numbers. Q = {p/q | p ∈ Z ∧ q ∈ Z ∧ q 6= 0}
CS 2233 Discrete Mathematical Structures
Sets and Set Structures – 3
Size of Sets
The cardinality of a set is the number of elements in the set.
Notation. |A| = n ≡ cardinality of A is n.
The number of elements in a finite set is a nonnegative integer. For
example, A = {x | x ∈ N ∧ x ≤ 100} has 101 elements.
If a set is not finite, then it is infinite.
Z and R are examples of infinite sets.
Theorem: Z+ is an infinite set.
Suppose Z+ is finite. Implies ∃n(|Z+| = n)
But A = {1, 2, . . . , n + 1} ⊆ Z+ , and |A| = n + 1, which is larger than n.
A subset can’t be larger, so a contradiction.
CS 2233 Discrete Mathematical Structures
1
Sets and Set Structures – 4
2
Sets of Sets
Set Identities
Power Sets
A set can have sets as elements, for example,
{{0, 1, 2, 3, 5, 8}, {1, 2, 4, 8}, {2, 3, 5, 7}}
The power set of a set is all of its subsets.
Power set of {1, 3} = {∅, {1}, {3}, {1, 3}}.
If |S| = n, then |power set of S| = 2n.
Set identities include De Morgan’s laws for sets.
A∪B =A∩B
A membership table can be used to prove set identities. It considers each
combination of sets an element might be in.
Tuples and Cartesian products
An n-tuple (a1 , . . . , an ) orders n elements.
A 2-tuple is called an ordered pair.
Note: {1, 3} = {3, 1}, but (1, 3) 6= (3, 1).
Cartesian product A×B = {(a, b) | a ∈ A ∧ b ∈ B}
{1, 3} × {2, 4} = {(1, 2), (1, 4), (3, 2), (3, 4)}
If |A| = m and |B| = n, then |A × B| = mn.
CS 2233 Discrete Mathematical Structures
A∩B =A∪B
A
1
1
0
0
B A∪B A∪B
1
1
0
0
1
0
1
1
0
0
0
1
A
0
0
1
1
B A∩B
0
0
1
0
0
0
1
1
CS 2233 Discrete Mathematical Structures
Sets and Set Structures – 7
Sets and Set Structures – 5
Venn Diagrams
Set Operations
A Venn diagram can be used to show the result of set operations.
Let U be the universal set.
union
A ∪ B = {x | x ∈ A ∨ x ∈ B}
intersection A ∩ B = {x | x ∈ A ∧ x ∈ B}
difference
A − B = {x | x ∈ A ∧ x 6∈ B}
complement A = {x | x ∈ U ∧ x 6∈ A}
A and B are disjoint ≡ A ∩ B = ∅
U
For a finite U , bit strings can represent sets.
U
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 1111111111
∅
{}
0000000000
A
{2, 3, 5, 7}
0011010100
B
{1, 3, 5, 7, 9}
0101010101
A ∩ B {3, 5, 7}
0001010100
CS 2233 Discrete Mathematical Structures
A
U
A
B
B
C
Sets and Set Structures – 6
CS 2233 Discrete Mathematical Structures
3
Sets and Set Structures – 8
4
Special Cardinalities
Let A and B be finite sets, where |A| = n and |B| = m.
|power set of A| = 2n
Each subset of A can be represented by a string of n bits. There are 2n
strings of n bits.
|A × B| = nm
Each element of A is paired with each element of B.
Principle of Inclusion-Exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|
|A| + |B| counts elements of |A ∩ B| twice.
CS 2233 Discrete Mathematical Structures
Sets and Set Structures – 9
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