Sets Prof. Richard Beigel Math C067 September 18, 2006 Revised September 20, 2006

Sets

Prof. Richard Beigel

Math C067

September 18, 2006

Revised September 20, 2006

What is a set?

• A set is a collection of zero or more objects– These objects are called elements

• No duplicates

• Order does not matter

Examples

• {2,3,5,7,11}

• {(1,1), (2,2), (3,3)}

• {Apple, Orange, Banana, Peach}

• {Apple, Dell, IBM}

• {Heads, Tails}

• {Win, Lose, Tie}

• {}

No Duplicates

• {1,1,2,3,5,8} is not a set

• {1,2,3,5,8} is a set

Order does not matter

• {3,4} = {4,3}

• {1,2,3,4,5} = {2,3,4,5,1}

• {1,4,2,5,3} = {1,3,5,2,4}

• {Apple, Dell, IBM} = {Dell, Apple, IBM}

Element-of Notation

• “x S” means that x is an element of the set S.

• 1 {1,2,3}

• 2 {1,2,3}

• 3 {1,2,3}

Not-an-element-of Notation

• “x S” means that x is not an element of the set S.

• 0 {1,2,3}

• 4 {1,2,3}

• 17 {1,2,3}

Universal Set (Universe)

• U contains all possible elements

Empty Set

• {} contains no elements at all

• {} = the set of 70-year-old Math C067 students

• {} = the set of 25-year-old U.S. Presidents is another symbol for the empty set

Set Descriptors

• The set consisting of all objects with some particular property can be denoted– {x : x has that particular property }

• {x : x is a positive integer less than 4} = {1,2,3}

• If it is understood that x is an integer we can write {x : 1 x < 4} = {1,2,3}

Examples with Letters

• Let U = {A,B,…,Z}• Let V = {x : x is a vowel} = {A,E,I,O,U}• Let C = {x : x is not a vowel} =

{B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z}

• Let G = {g : g is a grade} = {A,B,C,D,F} • Let P = {g : g is a passing grade} = {A,B,C,D}

Intersection

• The intersection of two sets A and B consists of all objects that belong to both A and B

• A B = {x : x A and x B}

• S {} = {} (true for every set S)

• S U = S (true for every set S)

Intersection Examples

• Let V = {x : x is a vowel} = {A,E,I,O,U}• Let C = {x : x is not a vowel} =


• Let G = {g : g is a grade} = {A,B,C,D,F} • Let P = {g : g is a passing grade} = {A,B,C,D}• P V = {A}

Venn Diagram for P V

C

D

B

OU

IE

A




• Let G = {g : g is a grade} = {A,B,C,D,F} • Let P = {g : g is a passing grade} = {A,B,C,D}• C P = {B,C,D}




• Let G = {g : g is a grade} = {A,B,C,D,F} • Let P = {g : g is a passing grade} = {A,B,C,D}• G P = {A,B,C,D} = P

Union

• The union of two sets A and B consists of all objects that belong to one or both sets A and B

• A B = {x : x A or x B}

• S {} = S (true for every set S)

• S U = U (true for every set S)

Union Examples

• Let U = {A,B,…,Z}

• Let V = {x : x is a vowel} = {A,E,I,O,U}

• Let C = {x : x is not a vowel} = {B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z}

• Let G = {g : g is a grade} = {A,B,C,D,F}

• Let P = {g : g is a passing grade} = {A,B,C,D}

• P V = {A,B,C,D,E,I,O,U}

Venn Diagram for P V

C

D

B

OU

IE

A

Union Examples

• Let U = {A,B,…,Z}





• G P = {A,B,C,D,F}

Union Examples

• Let U = {A,B,…,Z}





• V C = U

Disjoint Sets

• Two sets A and B are disjoint • A and B have no elements in common • A B = {}

Disjointness Examples

• Let U = {A,B,…,Z}





• V and C are disjoint

Disjointness Examples

• {H,P} and {I,B,M} are disjoint

Venn Diagram of Disjoint Sets

P

HB M

I

Subsets

• A is a subset of B (written A B) • Every element of A is an element of B • A B = A • A B = B

Subset Examples

• Let U = {A,B,…,Z}





• P G

Venn Diagram for P G

C

D

B

F

A

Subset Examples

• Let U = {A,B,…,Z}





• V U

Subset Examples

• Let U = {A,B,…,Z}





• {} V

Complement

• The complement of a set S consists of everything (in the universal set) that does not belong to S

• It is denoted Sc

• Sc = {x : x S}

• Uc = {}, {}c = U

• (Sc)c = S

Complementation Examples

• Let U = {A,B,…,Z}





• V = Cc

Complementation Examples

• Let U = {A,B,…,Z}





• C = Vc

Venn Diagram for Vc

B

U

E

O

I

HC D F G J PK L M N

Q WR S T V X Y Z

A

DeMorgan’s Laws

• (A B)c = Ac Bc

• (A B)c = Ac Bc

Captain Morgan’s Law

=

Proof of DeMorgan’s 1st Law

A B

U

A B

(A B)c

U

A B

Ac

U

A

Bc

U

B

Ac Bc

U

B

U

A

U

B

Proof of DeMorgan’s 2nd Law

• DeMorgan’s 1st Law says(A B)c = Ac Bc

• Apply DeMorgan’s 1st Law to Ac and Bc

(Ac Bc)c = (Ac)c (Bc)c

(Ac Bc)c = A B

• Complement both sides of the equation:((Ac Bc)c)

c = (A B)c

Ac Bc = (A B)c

Cardinality (Size) of Sets

• |S| = n(S) = the number of elements of the set S

Cardinality Examples

• |{2,3,5,7,11}| = 5

• |{(1,1), (2,2), (3,3)}| = 3

• |{Apple, Orange, Banana, Peach}| = 4

• |{Apple, Dell, IBM}| = 3

• |{Heads, Tails}| = 2

• |{Win, Lose, Tie}| = 3

• |{}| = 0


• Let U = {A,B,…,Z}





• |U| = 26


• Let U = {A,B,…,Z}





• |V| = 5


• Let U = {A,B,…,Z}





• |C| = 26-|V| = 26-5 = 21


• Let U = {A,B,…,Z}





• |G| = 5


• Let U = {A,B,…,Z}





• |P| = 4

Cardinality of Union

• |A B| =? |A| + |B|

• Let V = {A,E,I,O,U} 5

• Let P = {A,B,C,D} +4

• V P = {A,B,C,D,E,I,O,U} 8


• |A B| =? |A| + |B|

• Let V = {A,E,I,O,U} 5

• Let P = {A,B,C,D} +4

• V P = {A,B,C,D,E,I,O,U} 8


• |A B| = |A| + |B| |A B|

• Let V = {A,E,I,O,U} 5

• Let P = {A,B,C,D} +4

• V P = {A} 1• V P = {A,B,C,D,E,I,O,U} =8

Cardinality of Disjoint Union

• If A and B are disjoint sets then|A B| = |A| + |B|

• Why?• Because A B = {},• |A B| = |A| + |B| |A B|

= |A| + |B| |{}| = |A| + |B| 0 = |A| + |B|

Example

• If A and B are disjoint sets then

|A B| = |A| + |B|

• Let H = {H,P} 2

• Let I = {I,B,M} +3

• H I = {}

• H I = {H,P,I,B,M} =5

Difference of Two Sets

• The difference of two sets A and B consists of everything that belongs to A but does not belong to B.

• It is denoted A B• It is also denoted A \ B

• A B = A Bc

Examples of Set Difference

• Let V = {A,E,I,O,U}

• Let P = {A,B,C,D}

• V P = {E,I,O,U}

• P V = {B,C,D}

V = (VP) (VP)

C

D

B

OU

IE

A

PV VP VP

P = (PV) (PV)

C

D

B

OU

IE

A

PV PV VP

Cardinality of a Set Difference

• A = (AB) (AB)

• Because AB and AB are disjoint

|A| = |AB| + |AB|

• Rearranging,

|A| |AB| = |AB|

• so |AB| = |A| |AB|

|V P| = |V| |V P|

• Let V = {A,E,I,O,U} 5

• Let P = {A,B,C,D}

• V P = P V ={A} 1• V P = {E,I,O,U} =4

• P V = {B,C,D}

|P V| = |P| |P V|

• Let V = {A,E,I,O,U}

• Let P = {A,B,C,D} 4

• V P = P V ={A} 1• V P = {E,I,O,U}

• P V = {B,C,D} =3

(Cartesian) Product of Two Sets

• A B = {(a,b) : a A and b B}

• Let A = {egg roll, soup}

• Let B = {lo mein, chow mein, egg fu yung}

• A B =

{(egg roll,lo mein), (egg roll, chow mein),

(egg roll,egg fu yung), (soup,lo mein),

(soup,chow mein), (soup,egg fu yung)}

Cardinality of A B

• |A B| = |A| |B|(the raised dot means multiply)

• Let A = {egg roll, soup}

• Let B = {lo mein, chow mein, egg fu yung}

• |A B| = |A| |B| = 2 3 = 6

Another Example

• Let A = {1,2,3,4}

• Let B = {a,b,c}

• A B = {(1,a), (2,a), (3,a), (4,a), (1,b), (2,b),

(3,b), (4,b), (1,c), (2,c), (3,c), (4,c)}

= {(1,a), (2,a), (3,a), (4,a),

(1,b), (2,b), (3,b), (4,b),

(1,c), (2,c), (3,c), (4,c)}

• 43 rectangle contains 43 points

Longer Products

• A B C = (A B) C = A (B C)

= {(a,b,c) : a A and b B and c C}

• |A B C| = |A| |B| |C|

• A2 = A A

• A3 = A A A, etc.

• |Ak|= |A|k

Examples

• Let A = {Heads,Tails}

• A2 = {(Heads,Heads), (Heads,Tails)

(Tails,Heads), (Tails, Tails)}

• A3 = {(Heads,Heads,Heads),(Heads,Heads,Tails),

(Heads,Tails,Heads),(Heads,Tails,Tails),

(Tails,Heads,Heads),(Tails,Heads,Tails),

(Tails,Tails,Heads),(Tails,Tails,Tails)}

Examples

• Let A = {Heads,Tails}

• |A| = 2

• |A2| = |A|2 = 22 = 4

• |A3| = |A|3 = 23 = 8

• |A4| = |A|4 = 24 = 16

Application 1

• Yong’s Chinese restaurant offers 3 entrees– lo mein, chow mein, and egg fu yung

• each of which can be prepared with– chicken, beef, pork, shrimp, or no meat

• Yong’s menu includes 2 desserts:– ice cream or lychee nuts

Application 1 (continued)

• Let M = {chicken,beef,pork,shrimp,plain}• Let E = {lo mein, chow mein, egg fu yung}• A dinner order without dessert can be

represented as an ordered pair (m,e) where m M and e E. The set of all such dinner orders is

M E = {(m,e) : m M and e E}The number of possible dinner orders is

|M E| = |M| |E| = 5 3 = 15.

Application 1 (continued)

• Let M = {chicken,beef,pork,shrimp,plain}• Let E = {lo mein, chow mein, egg fu yung}• Let D = {ice cream, lychee nuts}• A dinner order including dessert can be

represented as an ordered triple (m,e,d) where m M, e E, and d D. The set of all such dinner orders is M E D = {(m,e,d) : m M and e E and d E}|M E D | = |M| |E| |D| = 5 3 2 = 30

Application 2

• The set of possible outcomes for a single coin flip is {H,T}, where H stands for heads and T stands for tails.

• We represent the results of several coin flips as an ordered tuple.

• The set of possible outcomes when one coin is flipped 10 times or when 10 distinguishable coins are flipped is {H,T}10.

• The number of possible outcomes is |{H,T}10| = |{H,T}|10 = 210 = 1024

Documents

Sets Prof. Richard Beigel Math C067 September 18, 2006 Revised September 20, 2006