Download - Set and Sets Operations
1Lecturer: Shaykhah
Set definitionSet is the fundamental discrete structure on
which all other discrete structures are built.
Sets are used to group objects together. Often, the objects in a set have similar properties.
A set is an unordered collection of objects.
The objects in a set are called the elements, or members, of the set
2Lecturer: Shaykhah
Some Important Sets The set of natural numbers: N = {0, 1, 2, 3, . . .}The set of integers: Z = {. . . ,−2,−1, 0, 1, 2, . . .}The set of positive integers: Z+ = {1, 2, 3, . . .}The set of fractions: Q = {0,½, –½, –5, 78/13,…} Q ={p/q | pЄ Z , qЄZ, and q≠0 }The set of Real: R = {–3/2,0,e,π2,sqrt(5),…}
3Lecturer: Shaykhah
Notation used to describe membership in sets a set A is a collection of elements. If x is an element of A, we write xA; If not: xA. xA Say: “x is a member of A” or “x is in A”. Note: Lowercase letters are used for elements, capitals for
sets. Two sets are equal if and only if they have the same
elements A= B : x( x A x B)
also Two sets A and B are equal if A B and B
A. So to show equality of sets A and B, show:
A B B A
4Lecturer: Shaykhah
How to describe a set?List all the members of a set, when this is
possible. We use a notation where all members of the set are listed between braces. { }
Example : {dog, cat, horse}Sometimes the brace notation is used to describe
a set without listing all its members. Some members of the set are listed, and then ellipses (. . .) are used when the general pattern of the elements is obvious.
Example: The set of positive integers less than 100 can be
denoted by {1, 2, 3, . . . , 99}.
5Lecturer: Shaykhah
How to describe a set?Another way to describe a set is to use set
builder notation. We characterize all those elements in the set by stating the property or properties they must have to be members.
Example: the set O of all odd positive integers less than
10 can be written as: O = {x | x is an odd positive integer <10}
or, specifying the universe as the set of positive integers, as O = {x Z+ | x is odd and x<10}.
6Lecturer: Shaykhah
Sets
The Empty Set (Null Set)The Empty Set (Null Set) We use to denote the empty set, i.e. the set with no
elements. For example: the set of all positive integers that are greater than their
squares is the null set.
Singleton setSingleton setA set with one element is called a singleton set.
7Lecturer: Shaykhah
Sets Computer ScienceComputer Science
Note that the concept of a datatype, or type, in computer science is built upon the concept of a set. In particular, a datatype is the name of a set, together with a set of operations that can be performed on objects from that set.
For example, Boolean is the name of the set {0, 1} together with operators on one or more elements of this set, such as AND, OR, and NOT.
8Lecturer: Shaykhah
Module #3 - Sets
04/20/23 (c)2001-2003, Michael P. Frank
Computer Representation of Sets
• Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The bit string (of length |U| = 10) that represents the set A = {1, 3, 5, 6, 9} has a one in the first, third, fifth, sixth, and ninth position, and zero elsewhere. It is
1 0 1 0 1 1 0 0 1 0.
SetsVenn diagramsVenn diagrams
Sets can be represented graphically using Venn diagrams.
In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle.
Inside this rectangle, circles or other geometrical figures are used to represent sets.
Sometimes points are used to represent the particular elements of the set.
10Lecturer: Shaykhah
SetsExample: A Venn diagram that represents V = {a, e, i,
o, u}, the set of vowels in the English alphabet
11Lecturer: Shaykhah
Subset The set A is said to be a subset of B if and only if every element of A is
also an element of B. We use the notation A B to indicate that A is a subset of the set B.
We see that A B if and only if the quantification x (x A → x B) is true.
12Lecturer: Shaykhah
SubsetsFor every set S,
1. S2. S S
Proper subset:When a set A is a subset of a set B but A ≠ B,
A B, and A B.We write A B and say that A is a proper subset of B
For A B to be true, it must be the case that x ((x A) (x B)) x ((x B) (x A))
13Lecturer: Shaykhah
SubsetsQuick examples:{1,2,3} {1,2,3,4,5}{1,2,3} {1,2,3,4,5}
Is {1,2,3}?Is {1,2,3}?Is {,1,2,3}?Is {,1,2,3}?
No !
Yes !
Yes !
Yes !Because to conclude it isn’t a subset we have to find an
element in the null set that is not in the set {1,2,3}. Which is not the case
14Lecturer: Shaykhah
SubsetsQuiz Time:
Is {x} {x,{x}}?
Is {x} {x,{x}}?
Is {x} {x}?
Is {x} {x}?
Yes !
Yes !
Yes !
No !
15Lecturer: Shaykhah
Finite and Infinite SetsFinite setFinite set
Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S.
The cardinality of S is denoted by |S|. |A B| = |A| + |B| - |A B| Infinite setInfinite set
A set is said to be infinite if it is not finite. For example, the set of positive integers is infinite.
16Lecturer: Shaykhah
CardinalityFindS = {1,2,3}, S = {3,3,3,3,3}, S = , S = { , {}, {,{}} },S = {0,1,2,3,…}, |S| is infinite
|S| = 3.
|S| = 1.
|S| = 0.
|S| = 3.
17Lecturer: Shaykhah
SetsWays to Define Sets:
Explicitly: {John, Paul, George, Ringo}
Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}
Set builder: { x : x is prime }, { x | x is odd }. In general { x :
P(x) is true }, where P(x) is some description of the set.
18Lecturer: Shaykhah
The power of a set Many problems involve testing all combinations of
elements of a set to see if they satisfy some property. To consider all such combinations of elements of a set
S, we build a new set that has as its members all the subsets of S.
Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S).
if a set has n elements , then the power has 2n elements
19Lecturer: Shaykhah
The power of a set Example:What is the power set of the set {0, 1, 2}? P({0,1,2}) is the set of all subsets of {0, 1, 2} P({0,1,2})= { , {0},{1},{2},{0,1},{0,2},
{1,2},{0,1,2}}
What is the power set of the empty set? What is the power set of the set {}
P()= {} P({})= {,{}}
N.B. the power set of any subset has at least two
elementsThe null set and the set
itself20Lecturer: Shaykhah
The Power SetQuick Quiz:Find the power set of the following:
S = {a},
S = {a,b},
S = ,
S = {,{}},
P(S)= {, {a}}.
P(S) = {, {a}, {b}, {a,b}}.
P(S) = {}.
P(S) = {, {}, {{}}, {,{}}}.
21Lecturer: Shaykhah
Cartesian Products The order of elements in a collection is
often important. Because sets are unordered, a different
structure is needed to represent ordered collections.
This is provided by ordered n-tuples. The ordered n-tuple (a1, a2, . . . , an) is the
ordered collection that has a1 as its first element, a2 as its second element, . . . , and an as its nth element.
22Lecturer: Shaykhah
Cartesian ProductsLet A and B be sets. The Cartesian product of
A and B, denoted by A×B, is the set of all ordered pairs (a, b), where aA and bB.
A×B = {(a, b) | a A b B}.
A1×A2×…×An=
{(a1, a2,…, an) | aiAi for i=1,2,…,n}.
A×B not equal to B×A
23Lecturer: Shaykhah
Cartesian ProductsExample:
What is the Cartesian product A × B × C, where
A = {0, 1}, B = {1, 2}, and C = {0, 1, 2}?Solution:AxBxC = {(0,1,0), (0,1,1), (0,1,2), (0,2,0),
(0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}
24Lecturer: Shaykhah
Notation with QuantifiersWhenever we wrote xP(x) or xP(x), we
specified the universe of using explicit English language
Now we can simplify things using set notation!
ExamplexR (x20) xZ (x2=1)Also mixing quantifiers:
a,b,cR xC(ax2+bx+c=0)25Lecturer: Shaykhah
Sets Operations
26Lecturer: Shaykhah
UNIONThe union of two sets A and B is:
A B = { x : x A v x B}If A = {1, 2, 3}, and B = {2, 4}, thenA B = {1,2,3,4}
AB
27Lecturer: Shaykhah
IntersectionThe intersection of two sets A and B is:
A B = { x : x A x B}If A = {Charlie, Lucy, Linus}, and B = {Lucy,
Desi}, thenA B = {Lucy}
AB
28Lecturer: Shaykhah
IntersectionIf A = {x : x is a US president}, and
B = {x : x is deceased}, then
A B = {x : x is a deceased US president}
AB
29Lecturer: Shaykhah
DisjointIf A = {x : x is a US president}, and B = {x : x is in this room},
then
A B = {x : x is a US president in this room} =
Sets whose intersection is empty are
called disjoint sets
AB
30Lecturer: Shaykhah
ComplementThe complement of a set A is:
A = A’ = { x : x A}If A = {x : x is bored}, thenA = {x : x is not bored} =
AU
= Uand
U =
A B = B A
31Lecturer: Shaykhah
Module #3 - Sets
04/20/23 (c)2001-2003, Michael P. Frank
Example
Let A and B are two subsets of a set E such that AB = {1, 2}, |A|= 3, |B| = 4, A = {3, 4, 5, 9} and B = {5, 7, 9}. Find the sets A, B and E.
E
A = {1, 2, 7}, B = {1, 2, 3, 4},
E = {1, 2, 3, 4, 5, 7, 9}
A 7 1 3 B 2 4 5 9
DifferenceThe set difference, A - B, is:
A - B = { x : x A x B }A - B = A B
A
U
B
33Lecturer: Shaykhah
Symmetric DifferenceThe symmetric difference, A B, is:
A B = { x : (x A x B) v (x B x
A)}
= (A - B) U (B - A)A
U
B
Like“ exclusive or”
34Lecturer: Shaykhah
Symmetric DifferenceExampleLet A = {1,2,3,4,5,6,7}
B = {3,4,p,q,r,s}Then we have
A U B = {1,2,3,4,5,6,7,p,q,r,s}A B = {3,4}
We getA B = {1,2,5,6,7,p,q,r,s}
35Lecturer: Shaykhah
Proving Set Equivalences• Recall that to prove such identity, we must
show that:1. The left-hand side is a subset of the right-hand
side2. The right-hand side is a subset of the left-hand
side3. Then conclude that the two sides are thus equal
• The book proves several of the standard set identities.
• We will give a couple of different examples here.
36Lecturer: Shaykhah
Proving Set Equivalences: Example A (1)Let
A={x|x is even} B={x|x is a multiple of 3}C={x|x is a multiple of 6}
Show that AB=C
37Lecturer: Shaykhah
Proving Set Equivalences: Example A (2)AB C: x AB
x is a multiple of 2 and x is a multiple of 3 we can write x=2.3.k for some integer k x=6k for some integer k x is a multiple of 6 x C
CAB: x C x is a multiple of 6 x =6k for some integer k x=2(3k)=3(2k) x is a multiple of 2 and of 3 x AB
38Lecturer: Shaykhah
Proving Set Equivalences: Example B (1)An alternative prove is to use membership
tables where an entry is1 if a chosen (but fixed) element is in the set0 otherwise
Example: Show that
A B C = A B C
39Lecturer: Shaykhah
Proving Set Equivalences: Example B (2)A B C ABC ABC A B C ABC
0 0 0 0 1 1 1 1 1
0 0 1 0 1 1 1 0 1
0 1 0 0 1 1 0 1 1
0 1 1 0 1 1 0 0 1
1 0 0 0 1 0 1 1 1
1 0 1 0 1 0 1 0 1
1 1 0 0 1 0 0 1 1
1 1 1 1 0 0 0 0 0
• 1 under a set indicates that an element is in the set
• If the columns are equivalent, we can conclude that indeed the two sets are equal
40Lecturer: Shaykhah
TABLE 1: Set Identities
Identity Name
A U = AA U = A Identity laws
A U U = UA =
Domination laws
A U A = AA A = A Idempotent laws
(A) = A Complementation laws
A U B = B U AA B = B A
Commutative laws
A U (B U C) = (A U B) U CA (B C) = (A B) C
Associative laws
A (B U C) = (A B)U(A C)A U(B C) = (A U B) (A U C)
Distributive laws
A U B = A BA B = A U B
De Morgan’s laws
A U (A B) = AA (A U B) = A
Absorption laws
A U A = UA A = Complement laws
41Lecturer: Shaykhah
Let’s proof one of the Identities Using a Membership Table
TABLE 2: A Membership Table for the Distributive Property
A B C B U C A (B U C)
A B A C (A B) U (A C)
1 1 1 1 1 1 1 1
1 1 0 1 1 1 0 1
1 0 1 1 1 0 1 1
1 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0
42Lecturer: Shaykhah
A (B U C) = (A B)U(A C)
ANY QUESTIONS???Refer to chapter 2 of the book for further
reading
43Lecturer: Shaykhah