Set Builder Notation - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

CS 2610

September 4, 2008

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Agenda

Set Theory  Set Builder Notation  Universal Set  Power Set and Cardinality  Set Operations  Set Identities  Cartesian Product

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Universal Set

Universal Set is the set containing all the objects under consideration.

It is denoted by U

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Set Builder Notation

Set Builder – characterize the elements in a set by stating the properties that the elements must have to belong to the set.

{ x | P (x) }

 reads x that satisfy P(x), x such that P(x)  x belongs to a universal set U.

concise definition of a set

Examples:

P = { x | x is prime number} U : Z + M={ x | x is a mammal} U: All animals Q+^ = { x ∈ R | x = p/q, for some positive integers p, q }

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Subsets

A ⊆ B means “A is a subset of B” or, “B contains A”

“every element of A is also in B” or, ∀x ((x ∈ A) → (x ∈ B))

A ⊆ B means “A is a subset of B” B ⊇ A means “B is a superset of A”

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Subsets

A ⊆ B means “A is a subset of B”

For Every Set S,

i) ∅ ⊆ S the empty set is a subset of every set

ii) S ⊆ S every set is a subset of itself

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Set Equality

A = B if and only if A and B have exactly the same elements.

iff, A ⊆ B and B ⊆ A iff, A ⊆ B and A ⊇ B iff, ∀x ((x ∈ A) ↔ (x ∈ B)).

 To show equality of sets A and B, must prove both:

• A ⊆ B
• B ⊆ A

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Set Cardinality

The cardinality of a set is the number of distinct

elements in the set. |S | denotes the cardinality

of S.

S = {1,2,3}

S = {5,5,5,5,5,5}

S = ∅

S = { ∅, {∅}, {∅,{∅}} }

A set S is said to be finite if its cardinality is a

nonnegative integer. Otherwise, S is said to be

infinite.

Given N = {0,1,2,3,…}, | N | is infinite (natural nos.)

|S| = 3

|S| = 1

|S| = 0

|S| = 3

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n-Tuples

An ordered n- tuple, n ∈ Z + , is an ordered list

(a 1 , a 2 , …, an ).

 Its first element is a 1.  Its second element is a 2 , etc.  Enclosed between parentheses (list not set).

Order and length matters:

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Cartesian Product

The Cartesian Product of two sets A and B is:

A x B = { (a, b) | a ∈ A ∧ b ∈ B}

Example: A= {a, b}, B= {1, 2} A × B = {(a,1), (a,2), (b,1), (b,2)} B × A = {(1,a), (1,b), (2,a), (2,b)} Not commutative!

In general, A 1 x A 2 x … x A (^) n = {(a 1 , a 2 ,…, an) | a 1 ∈ A 1 , a 2 ∈ A 2 , …, an ∈ An} |A 1 x A 2 x … x A (^) n| = |A 1 | x |A 2 | x … x |A (^) n|

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Intersection Operator

The intersection of two sets A and B is:

A ∩ B = { x | x ∈ A ∧ x ∈ B}

Example: A = {1,2,3}, B = {1,6} A ∩ B = {1}

Two sets A, B are called disjoint iff their

intersection is empty.

A ∩ B = ∅

Example: A = {1,2,3}, B = {9,10}, C = {2, 9} A and B are disjoint sets, but A and C are not

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Set Theory : Inclusion/Exclusion

What is the cardinality of A ∪ B?

A A∩ B B

Once

twice

|A∪B| = |A| + |B| - |A ∩ B|

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Set Difference

The set difference, A - B, is:

A - B = { x | x ∈ A ∧ x ∉ B }

Example:

A = {2,3,4,5 }, B = {3,4,7,9 }

A- B = {2, 5}

B – A = {7,9}

It is not commutative!!