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


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.


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))


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


SubsetsQuiz Time:

Is {x} {x,{x}}?

Is {x} {x,{x}}?

Is {x} {x}?

Is {x} {x}?

Yes !

Yes !

Yes !

No !


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.


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.


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.


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


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) = {, {}, {{}}, {,{}}}.


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.


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


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)}


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


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


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


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


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


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


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


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”


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}


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.


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


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


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


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


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


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


A (B U C) = (A B)U(A C)

ANY QUESTIONS???Refer to chapter 2 of the book for further

reading


set and sets operations

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