Set Theory - Discrete Structures - Lecture Slides, Slides of Discrete Structures and Graph Theory

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… and now for something completely

different…

• Set Theory

Actually, you will see that logic and

set theory are very closely related.

Set Theory

• Set: Collection of objects (“elements”)
• a∈A “a is an element of A” “a is a member of A”
• a∉A “a is not an element of A”
• A = {a 1 , a 2 , …, a (^) n } “A contains…”
• Order of elements is meaningless
• It does not matter how often the same element is listed.

Examples for Sets

• “Standard” Sets:
• Natural numbers N = {0, 1, 2, 3, …}
• Integers Z = {…, -2, -1, 0, 1, 2, …}
• Positive Integers Z+^ = {1, 2, 3, 4, …}
• Real Numbers R = {47.3, -12, π, …}
• Rational Numbers Q = {1.5, 2.6, -3.8, 15, …} (correct definition will follow)

Examples for Sets

• A = ∅ “empty set/null set”
• A = {z} Note: z∈A, but z ≠ {z}
• A = {{b, c}, {c, x, d}}
• A = {{x, y}} Note: {x, y} ∈A, but {x, y} ≠ {{x, y}}
• A = {x | P(x)} “set of all x such that P(x)”
• A = {x | x∈ N ∧ x > 7} = {8, 9, 10, …}

“set builder notation”

Subsets

• A ⊆ B “A is a subset of B”
• A ⊆ B if and only if every element of A is also an element of B.
• We can completely formalize this:
• A ⊆ B ⇔ ∀x (x∈A → x∈B)
• Examples:

A = {3, 9}, B = {5, 9, 1, 3}, A ⊆ B? true

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A ⊆ B?

false

true

A = {1, 2, 3}, B = {2, 3, 4}, A ⊆ B?

Subsets

• Useful rules:
• A = B ⇔ (A ⊆ B) ∧ (B ⊆ A)
• (A ⊆ B) ∧ (B ⊆ C) ⇒ A ⊆ C (see Venn Diagram)

U

A

B

C

Cardinality of Sets

• If a set S contains n distinct elements, n∈ N , we call S a finite set with cardinality n.
• Examples:
• A = {Mercedes, BMW, Porsche}, |A| = 3

B = {1, {2, 3}, {4, 5}, 6} |B| = 4

C = ∅ |C| = 0

D = { x∈ N | x ≤ 7000 } |D| = 7001

E = { x∈ N | x ≥ 7000 } E is infinite!

The Power Set

• P(A) “power set of A”
• P(A) = {B | B ⊆ A} (contains all subsets of A)
• Examples:
• A = {x, y, z}
• P(A) = {∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}
• A = ∅
• P(A) = {∅}
• Note: |A| = 0, |P(A)| = 1

Cartesian Product

• The ordered n-tuple (a 1 , a 2 , a 3 , …, a (^) n ) is an ordered

collection of objects.

• Two ordered n-tuples (a 1 , a 2 , a 3 , …, a (^) n ) and

(b 1 , b 2 , b 3 , …, bn ) are equal if and only if they contain

exactly the same elements in the same order, i.e. a (^) i = bi

for 1 ≤ i ≤ n.

• The Cartesian product of two sets is defined as:
• A×B = {(a, b) | a∈A ∧ b∈B}
• Example: A = {x, y}, B = {a, b, c}

A×B = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

Cartesian Product

• The Cartesian product of two sets is defined as: A×B =

{(a, b) | a∈A ∧ b∈B}

• Example:
• A = {good, bad}, B = {student, prof}
• A×B = { (^) (good, student), (good, prof), (bad, student), (bad, prof)}

B×A = { (student, good), (prof, good), (student, bad), (prof, bad)}

Set Operations

• Union: A∪B = {x | x∈A ∨ x∈B}
• Example: A = {a, b}, B = {b, c, d}
• A∪B = {a, b, c, d}
• Intersection: A∩B = {x | x∈A ∧ x∈B}
• Example: A = {a, b}, B = {b, c, d}
• A∩B = {b}

Set Operations

• Two sets are called disjoint if their intersection is empty, that is, they share no elements:
• A∩B = ∅
• The difference between two sets A and B contains exactly those elements of A that are not in B:
• A-B = {x | x∈A ∧ x∉B} Example: A = {a, b}, B = {b, c, d}, A-B = {a}

Set Operations

• Table 1 in Section 1.5 shows many useful equations.
• How can we prove A∪(B∩C) = (A∪B)∩(A∪C)?
• Method I:
• x∈A∪(B∩C)

⇔ x∈A ∨ x∈(B∩C)

⇔ x∈A ∨ (x∈B ∧ x∈C)

⇔ (x∈A ∨ x∈B) ∧ (x∈A ∨ x∈C) (distributive law for logical expressions)

⇔ x∈(A∪B) ∧ x∈(A∪C)

⇔ x∈(A∪B)∩(A∪C)

Set Operations

• Method II: Membership table
• 1 means “x is an element of this set” 0 means “x is not an element of this set”

1 1 1 1 1 1 1 1

1 1 0 0 1 1 1 1

1 0 1 0 1 1 1 1

1 0 0 0 1 1 1 1

0 1 1 1 1 1 1 1

0 1 0 0 0 1 0 0

0 0 1 0 0 0 1 0

0 0 0 0 0 0 0 0

A B C B∩C A∪(B∩C) A∪B A∪C (A∪B) ∩(A∪C)