Chapter 2: Concept of Set

1. SET

What is a Set?

A set is a collection of discrete data items. The members of the set can be numbers or

names.

WAY OF LISTING THE ELEMENTS OF SETS

(i) ROSTER METHOD

-list all the members of a set, when this is possible.

-use a notation where all members of the set are listed between braces.

Example: The set K of odd positive integers less than 10 can be expressed by K = {1, 3, 5, 7, 9}.

(ii)SET BUILDER NOTATION

-characterize all those elements in the set by stating the property they must have to be members of the set.

Example:   

B = {y | y is an odd positive integer less than 10},

                     OR

specifying the universe as the set of positive integers, as :

B = {y ∈ Z+ | y is odd and y < 10}.

IMPORTANT !

N = {0, 1, 2, 3, . . .}, the set of natural numbers

Z = {. . . ,−2,−1, 0, 1, 2, . . .}, the set of integers

Z+ = {1, 2, 3, . . .}, the set of positive integers

Q = {p/q | p ∈ Z, q ∈ Z, and q _= 0}, the set of rational numbers

R, the set of real numbers

R+, the set of positive real numbers

C, the set of complex numbers.

Two sets are equal if and only if they have the same elements.

Example:set Y= {2, 4, 6}

        set Z={6, 2, 4}

Y and Z are equal if an only if ∀x(x ∈ Y ↔ x ∈ Z).

The way the elements in the sets are listed does not matter.

2. SPECIFYING PROPERTIES OF SETS

It is inconvenient, and sometimes impossible, to define a set by listing all its elements

Alternatively, we can define by a set by describing the properties that its elements should satisfy.

We use the notation     

{  x ϵ A | P(x)}

to define the set as the set of elements, x, in A such that x satisfies property P.

e.g.

 {  x | x is a prime number and x  < 1000 }

 {  x | x is a real number and  -2 < x < 5 }

3. SET MEMBERSHIP

Set membership is when one set is an element of another.

Examples:

7 ∈ X, symbolizes 7 is an element of X .

Z  = { A , R , T , I , F , C , L }

Alphabet A is one of the alphabets contained in the word “ARTIFICIAL”. We can say that alphabet A and I is an element of set Z, or in the mathematical form: A ∈ Z , I ∈ Z .

* set is any collection of object .

* the object is called elements of the set.

* the simplest way is by listing the element enclose between curly brackets or 'braces' {} .

4. EMPTY SET

Set D is a null set. 

To show that set D is a null set, which is also an empty set. We write the set in mathematical form like below: 

D = {} OR D = ∅ 

5. SET OF NUMBER

Definition- All the numbers in the Number System are classified into different sets and those sets are called as Number Sets.

The set of real numbers is divided into natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Examples of number set

• N= {1, 2, 3…….} is the set of natural numbers.
• (0, 1, 2, 3……} is the set of whole numbers.
• Z= {……-3, -2, -1, 0, 1, 2, 3….} is the set of integers.
• Q=The numbers 2/3,3/5,10.3, 102.25, etc. are rational numbers but not integers.
• √2=1.414213, √5=2.236067 , etc. are irrational numbers.

N - Natural number

Z – Integers

Q - Rational numbers

A – Algebraic numbers

R – Real numbers

I – Imaginary numbers, Numbers that when squared give a negative result.

C – Complex numbers, A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary.

Illustration Natural numbers are a subset of Integers Integers are a subset of Rational Numbers Rational Numbers are a subset of the Real Numbers Combinations of Real and Imaginary numbers make up the Complex Numbers.

6. SET EQUALITY

Equality of sets is defined as set A is said to be equal to set B if both sets have the same elements or members of the sets, i.e. if each element of set A is also belongs to each element of set B as well as each element of set B is also belongs to each element of set A.
          
It can be written as A Ì B and B Ì A in this case we write as.A = B If there is at least one elements of B which is not in A, then A is not equal to B and we write A ≠ B .

Example:

Two sets are equal if they contain the same identical elements. If two sets have only the same number of elements, then the two sets are One-to-One correspondence. Equal sets are One-to-One correspondence but correspondence sets are not always equal sets.

Ex: A: (1, 2, 3, 4)
B: (h, t, m, k)
C: (4, 1, 3, 2)

A and C are Equal sets and 1-1 correspondence sets.

7. VENN DIAGRAM

   FIGURE 1 :    Venn Diagram showing that Y is a subset of X

8. SUBSETS

A Í B           “A is a subset of B”

A Í B if and only if every element of A is also 
          an element of B.

We can completely formalize this:

A Í B Û "x (xÎA ® xÎB)

Useful rules:

•       A = B Û (A Í B) Ù (B Í A)

•       (A Í B) Ù (B Í C) Þ A Í C   (see Venn Diagram)

 

Proper subsets:

A Ì B     “A is a proper subset of B”

A Ì B Û "x (xÎA ® xÎB) Ù $x (xÎB Ù xÏA)

or

A Ì B Û "x (xÎA ® xÎB) Ù Ø"x (xÎB ® xÎA)

9. POWER SETS

power set :   

       pow(A) : : = {S|S Í A}

In words, the power set pow(A) of a set A 

contains all the subsets of A as members

OR

Power set

of S is the set of all subsets, denoted by P( S )

The power set of A has 2|A| elements.

Example

A = {1, 2, 3}

| A | = 3

P( A ) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

| P( A ) | = 2|A| = 23 = 8

10. SET OPERATIONS

Union: AÈB = {x | xÎA Ú xÎB}

Example: A = {a, b}, B = {b, c, d}

                 AÈB = {a, b, c, d}

Intersection: AÇB = {x | xÎA Ù xÎB}

Example: A = {a, b}, B = {b, c, d}

                  AÇB = {b}

Two sets are called disjoint if their intersection is empty and  they share no elements:

AÇB = Æ

The difference between two sets A and B contains exactly those elements of A that are not in B:

A-B = {x | xÎA Ù xÏB}
Example: A = {a, b}, B = {b, c, d}, A-B = {a}

The complement of a set A contains exactly those elements under consideration that are not in A:

-A = U-A

Example: U = N,  B = {350, 351, 352, …}

                         -B = {0, 1, 2, …, 348, 349}

Characteristics of Sets

11.  GENERALISED UNION AND INTERSECTION

Union

of a collection of sets contains elements that are members of at least one set in the collection.

Intersection

of a collection of sets contains elements that are members of all sets in the collection.

Example

A = {0, 2, 4, 6, 8}

B = {0, 1, 2, 3, 4}

C = {0, 1, 3, 5, 7}

      A ∩ B ∩ C = { 0 }

      A ∪B ∪ C = { 0, 1, 2, 3, 4, 5, 6, 7, 8 }

12. CARTESIAN PRODUCT

-DEFINITION

The Cartesian product is the direct product of the elements of two sets.

Example 1 : If X and Y are two non empty sets, then the Cartesian product is defined as follows:

X x Y = {(x , y) | x ∈ X and y ∈ Y}

Example 2 : Let A = {1,2,3} and B={a, b, c} Then A x B={(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)}

       

-ORDERED PAIR

Ordered  pairs are a set having two elements, where the numbers are written in a particular order. The ordered pair with the first element x and second element y is written  as (x, y). 

-CARTESIAN JOIN

A Cartesian Join is another name for a Cartesian product. Cartesian Join of two sets A and B is the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B.

Example 1: Let A={1,2,3} B={a, b} C={*}

Then Cartesian Join A x B x C = {(1,a, *) (1,b, *), (2,a, *) (2,b, *), (3,a, *) (3,b,*)}

-PROPERTIES OF CARTESIAN PRODUCT

1) A x B = B x A =>A = B (A and B are equal Sets)

2) Cartesian product is not Commutative (except for empty Sets) A x B ≠ B x A 3) Cartesian Product of Empty Set

The Cartesian product of any set A with the empty set ∅ or { } is an empty set, since we cannot form any ordered pairs. A x ∅ = ∅ x A = ∅

Examples on Cartesian Product of Empty Set

Example 1: If A = {a, b, c} and B = { } then Cartesian Product of A x B = { }, since set B is an empty set. 4) Cartesian product is not Associative   (A x B) x C ≠ A x (B x C) 5) (A ∩ B) x (C ∩ D) = (A x C) ∩ (B x D) 6) (A ∪ B) x (C ∪ D) = (A x C) ∪ (B x D) 7) A x (B ∩ C) = (A x B) ∩ (A x C) 8) A x (B ∪ C) = (A x B) ∪ (A x C)