University of Hawaii ICS141: Discrete Mathematics for Computer Science Ijanst/141/lecture/08-Sets1.pdf · ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer

8-1 ICS 141: Discrete Mathematics I- Fall 2011

University of Hawaii

ICS141: Discrete Mathematics for Computer Science I

Dept. Information & Computer Sci., University of Hawaii

Jan Stelovsky based on slides by Dr. Baek and Dr. Still

Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by McGraw-Hill

8-2 ICS 141: Discrete Mathematics I- Fall 2011


Lecture 8

Chapter 2. Basic Structures 2.1 Sets 2.2 Set Operations

8-3 ICS 141: Discrete Mathematics I - Fall 2011

University of Hawaii Sets so far… n  Previously…

n  Literal set {a,b,c} and set-builder notation {x |P(x)} n  Basic properties: unordered, distinct elements n  infinite sets N, Z, Z+, Q, R

n  Today n  ∈ relational operator, and the empty set ∅ n  Venn diagrams n  Set relations =, ⊆, ⊆, ⊂, ⊃, ⊄, etc. n  Cardinality |S| of a set S n  Power sets P(S) n  Cartesian product S × T n  Set operators: ∪, ∩, -


University of Hawaii Basic Set Relations: Member of n  x∈S (“x is in S”) is the proposition that object x

is an ∈lement or member of set S.

n  e.g. 3 ∈ N, a ∈ {x | x is a letter of the alphabet}

n  Can define set equality in terms of ∈ relation: ∀S,T: S = T ↔ [∀x (x∈S ↔ x∈T)] “Two sets are equal iff they have all the same members.”

n  x∉S ≡ ¬(x∈S) “x is not in S”


University of Hawaii The Empty Set

n  ∅ (“null”, “the empty set”) is the unique set that contains no elements whatsoever.

n  ∅ = { } = {x | False}

n  No matter the domain of discourse, we have the axiom ¬∃x: x∈∅.

n  { } ≠ {∅} = { { } } n  {∅} it isn’t empty because it has ∅ as a member!


University of Hawaii Venn Diagrams U


University of Hawaii Subset and Superset n  S⊆T (“S is a subset of T ”) means that every

element of S is also an element of T. n  S⊆T ≡ ∀x (x∈S → x∈T) n  ∅⊆S, S⊆S n  S⊆T (“S is a superset of T ”) means T⊆S n  Note (S = T) ≡ (S⊆T ∧ S⊆T)

≡ ∀x(x∈S → x∈T) ∧ ∀x(x∈T → x∈S)

≡ ∀x(x∈S ↔ x∈T) n  S⊆T means ¬(S⊆T), i.e. ∃x(x∈S ∧ x∉T)


University of Hawaii Proper (Strict) Subsets & Supersets n  S⊂T (“S is a proper subset of T ”) means that

S⊆T but T⊆S. Similar for S ⊃T. n  Example:

{1, 2} ⊂ {1, 2, 3}

S T

Venn Diagram of S⊂T


University of Hawaii Sets Are Objects, Too! n  The objects that are elements of a set may

themselves be sets. n  Example:

Let S = {x | x ⊆ {1, 2, 3}} then S = { ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

n  Note that 1 ≠ {1} ≠ {{1}} !!!!


University of Hawaii Cardinality and Finiteness n  |S| (read “the cardinality of S”) is a measure

of how many different elements S has.

n  E.g., |∅| = 0, | {1, 2, 3} | = 3, | {a, b} | = 2,

| { {1, 2, 3}, {4, 5} } | = ____

n  If |S| ∈ N, then we say S is finite. Otherwise, we say S is infinite.

n  What are some infinite sets we’ve seen?

N, Z, Q, R


University of Hawaii The Power Set Operation n  The power set P(S) of a set S is the set of all

subsets of S. P(S) = {x | x⊆S}. n  Examples

n  P({a, b}) = { ∅, {a}, {b}, {a, b} } n  S = {0, 1, 2}

P(S) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} n  P(∅) = {∅} n  P({∅}) = {∅, {∅}}

n  Note that for finite S, |P(S)| = 2|S|. n  It turns out ∀S (|P(S)| > |S|), e.g. |P(N)| > |N|.


University of Hawaii Ordered n-tuples

n  These are like sets, except that duplicates matter, and the order makes a difference.

n  For n∈N, an ordered n-tuple or a sequence or list of length n is written (a1, a2, …, an). Its first element is a1, its second element is a2, etc.

n  Note that (1, 2) ≠ (2, 1) ≠ (2, 1, 1).

n  Empty sequence, singlets, pairs, triples, quadruples, quintuples, …, n-tuples.

Contrast with sets’ {}


University of Hawaii Cartesian Products of Sets n  For sets A and B, their Cartesian product

denoted by A × B, is the set of all ordered pairs (a, b), where a∈A and b∈B. Hence, A × B = { (a, b) | a∈A ∧ b∈B }.

n  E.g. {a, b} × {1, 2} = { (a, 1), (a, 2), (b, 1), (b, 2) }

n  Note that for finite A, B, |A × B| = |A||B|. n  Note that the Cartesian product is not

commutative: i.e., ¬∀A,B (A × B = B × A). n  Extends to A1 × A2 × … × An

= {(a1, a2, …, an) | ai ∈ Ai for i = 1, 2,…, n}


University of Hawaii The Union Operator

n  For sets A and B, their union A∪B is the set containing all elements that are either in A, or (“∨”) in B (or, of course, in both).

n  Formally, ∀A,B: A∪B = {x | x∈A ∨ x∈B}.

n  Note that A∪B is a superset of both A and B (in fact, it is the smallest such superset):

∀A,B: (A∪B ⊆ A) ∧ (A∪B ⊆ B)



n  {a, b, c} ∪ {2, 3} = {a, b, c, 2, 3} n  {2, 3, 5} ∪ {3, 5, 7} = {2, 3, 5, 3, 5, 7} = {2, 3, 5, 7}

Union Examples


University of Hawaii The Intersection Operator

n  For sets A and B, their intersection A∩B is the set containing all elements that are simultaneously in A and (“∧”) in B.

n  Formally, ∀A,B: A∩B = {x | x∈A ∧ x∈B}.

n  Note that A∩B is a subset of both A and B (in fact it is the largest such subset):

∀A,B: (A∩B ⊆ A) ∧ (A∩B ⊆ B)



n  {a, b, c} ∩ {2, 3} = ___

n  {2, 4, 6} ∩ {3, 4, 5} = ____

Intersection Examples

Think “The intersection of University Ave. and Dole St. is just that part of the road surface that lies on both streets.”

∅

{4}


University of Hawaii Disjointedness

n  Two sets A, B are called disjoint (i.e., unjoined) iff their intersection is empty. (A ∩ B = ∅)

n  Example: the set of even integers is disjoint with the set of odd integers.

Help, I’ve been

disjointed!


University of Hawaii Inclusion-Exclusion Principle n  How many elements are in A∪B? |A∪B| = |A| + |B| - |A∩B|

n  Example: How many students in the class major in Computer Science or Mathematics? n  Consider set E = C ∪ M,

C = {s | s is a Computer Science major} M = {s | s is a Mathematics major}

n  Some students are joint majors! |E| = |C ∪ M| = |C| + |M| - |C ∩ M|


University of Hawaii Inclusion-Exclusion Principle n  How many elements are in A∪B? |A∪B| = |A| + |B| - |A∩B|

n  Example: How many students in the class major in Computer Science or Mathematics? n  Consider set E = C ∪ M,

C = {s | s is a Computer Science major} M = {s | s is a Mathematics major}

n  Some students are joint majors! |E| = |C ∪ M| = |C| + |M| - |C ∩ M|


University of Hawaii Set Difference

n  For sets A and B, the difference of A and B, written A - B, is the set of all elements that are in A but not B.

n  Formally:

A - B = {x | x∈A ∧ x∉B} = {x | ¬(x∈A → x∈B)}

n  Also called: The complement of B with respect to A.


University of Hawaii Set Difference: Venn Diagram

n  A − B is what’s left after B “takes a bite out of A”

Set A

Set A - B

Set B

Chomp!


University of Hawaii Set Difference Examples

n  {1, 2, 3, 4, 5, 6} - {2, 3, 5, 7, 9, 11} =

___________

n  Z - N = {… , −1, 0, 1, 2, … } - {0, 1, … } = {x | x is an integer but not a natural #} = {… , −3, −2, −1} = {x | x is a negative integer}

{1, 4, 6}


University of Hawaii Set Complements n  The universe of discourse (or the domain)

can itself be considered a set, call it U.

n  When the context clearly defines U, we say that for any set A ⊆ U, the complement of A, written as , is the complement of A with respect to U, i.e., it is U - A.

n  E.g., If U = N,

A

{3,5} = {0,1, 2, 4, 6, 7,...}


University of Hawaii More on Set Complements

n  An equivalent definition, when U is obvious: }|{ AxxA ∉=

U

A

AA


University of Hawaii Interval Notation n  a, b ∈ R, and a < b then

n  (a, b) = {x ∈ R | a < x < b}

n  [a, b] = {x ∈ R | a ≤ x ≤ b}

n  (a, b] = {x ∈ R | a < x ≤ b}

n  (–∞, b] = {x ∈ R | x ≤ b}

n  [a, ∞) = {x ∈ R | a ≤ x}

n  (a, ∞) = {x ∈ R | a < x}

Documents

University of Hawaii ICS141: Discrete Mathematics for Computer Science Ijanst/141/lecture/08-Sets1.pdf · ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer