1.Sets and their Representations,Types of Sets

Chapter 1

SETS

Who is the founder of set theory?

Georg Cantor, in full Georg Ferdinand Ludwig Philipp Cantor, (born March 3, 1845, St. Petersburg, Russia—died January 6, 1918, Halle, Germany), German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.

Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline.

SET THEORY

Definition: Set is a collection of well defined or  well-determined objects or elements or numbers or members or functions .

Well defined means known characteristics and properties clearly expressed, explained, or described.

N.B.:-

(i) Sets, are generally denoted by capital English alphabets like A, B, C, P, Q, R etc.

(ii) Objects,elements,members of a set synonymous terms or similar or homogeneous terms.

(iii) The elements of a set are represented by small letters a,b,c,x,y,z, etc..

(iv) If a is an element of a set A, then a is belongs to A.   i.e., a ∈  A 

(v) If b is not an element of a set A, then b does not belong to A   i.e., b ∉   A

Examples:     

(i) The collection of three boys: Mohan, Ram, Prakash.

(ii) The collection N of all natural numbers.

(iii) The collection R of all real numbers.

(iv) The collection of all first year students of DAV

(v)  The collection of the roots of the equation x²⁷-1=0

But the following collections are not sets.

(i)  The collection of some natural numbers.

(ii) The collection of the politicians of India.

(iii) Collection of good cricket players of India.

These are not sets because the terms ‘some’ in (i) politicians in (ii) and good cricket players in (iii) are not well defined.

Example: Collection of tall students in your school.

It is not a set because tall is not well defined.

Similarly , 

Interesting, honesty, intelligence, better, tall ,etc… are not well defined.

Notations: -

• N: the set of natural numbers
• W: the set of Whole numbers
• I=Z: the set of Integer numbers
• Z⁺: the set of Positive Integer numbers
• Z^- : the set of negative Integer numbers
• Q : the set of Rational numbers
• Q’=T : the set of Irrational numbers
• C: the set of Complex numbers
• ∃ : there exists
• ∄ : there does not exists

Description or Representation of a set:  A set is generally described in two ways.

(1) Roster method / Tabular method / set by extension:

In this method the elements of the set are written within the curly brackets ({  }) and separated by a comma (,).

To write a set in Roster method elements are not to be repeated.

Example:        

(i) Set of all vowels of English alphabets.      =          {a, e, i, o u}

(ii) Set of all primes less than 8 =  {2, 3, 5, 7}

(2) Set Builder Method / Set selector method / method of specification:

It is represented by a symbol or variable that all the elements  of set possess a single common property and within the curly bracket which is not possessed by any elements outside the set.         

i.e., a set is described by a charactering properly of its elements.

Example: (i) The set of all even natural numbers is written by

A = {x|  x = 2n for n Î N}

(ii) The set of all prime numbers in between 5 and 15 is written by

B = { x|  x is a prime no. & 5 < x < 15}

Note:   If an element a  remains in a set A, we say etc. a belongs to A and can be written by x Î A. If a does not remain in A we say a does not belongs to A and is written by x Ï A.

Types of sets:

[1] Empty set / Null set / Void set / Phi set (ɸ):

A set is said to be empty set or null set or void set if it has no element. i.e., ɸ={ }

Example:

(i)  The set of all the months which are of 35 days.

(ii)   {x² | x² < 0}

[2] Singleton Set:

A set consisting of a single element is called a singleton set.

Example:

(i) A= { 5 }

(ii)  B= {Rama}

(iii) Present prime minister of India etc.

[3] Finite set:

A set is said to be finite if it is void set or its elements can be counted by natural numbers for any natural number. n Î N and the counting can be terminated for some n.

Example:

(i)  Set of even prime numbers.

(ii) {Arun, Mohan, Ramesh}

(iii) { 1, 2, 3, 4, -------- 100}

[4] Cardinality of a finite set:

The cardinality or cardinal number of a finite set is defined as the number of elements present in that set.

Cardinality of a set is denoted by n (A) on |A|

[5] Infinite sets:

A set whose elements cannot be listed by the natural numbers for any natural no. nÎN.

Example: A= set of points on a line.

N.B.:- All infinite sets cannot be described in the roster method.

[6] Equivalent sets:

Two finite sets A & B are said to be equivalent if their cardinal numbers are same i.e. n (A) = n (B)

Example: A = {1, 2, 3, 4, 5},  B = {a, b, c, d, e)

[7] Equal sets:

Two sets A and B are said to be equal if every elements of A is an element of B.

Example: {0, 1, 2, 3} = {2, 1, 0, 3}

[8] Subsets & super sets:

Let A and B be two sets such that every element of A is also an element of B, then A is said to be the subset of B and is denoted by AÌ B. and B is called the super set of A and denoted by B É A.

If n(A)=n, then Total No. of Subset of set A = 2ⁿ

Example:

            A = { a, e, i, o, u}

            B = {a, b, c, d, e, --------- x, y, z}

            Hence A Ì B & B É A

Theorem: The empty set (or void set) is a subset of every set. Empty set is represented by f.

Proof: Let A be any set and f must be a member of A .

            Þ  f Í A

Theorem:

The total number of subsets of a finite set containing n elements is 2ⁿ.

Proof: A be a finite set & |A| = n

Consider those subsets of A that have r elements. Since no. of selections or r elements from n different things is

Hence the total no. of subsets is

[9] Proper subset / proper set :

If A Ì B and B contain at least one element that is not in A, then A is called a proper subset of B.

Only itself is not Proper subset.

If n(A)=n, then Total No. of Proper Subset of set A = 2ⁿ-1

Example: A = {1, 2, 3}     B = {1, 2, 3, 4, 5, 6, 7}

Properties : (i) f is subset of every set’

                   (ii) If AÍ B and BÍ A ,then A=B

                   (iii) If A Í B and B Í C then A Í C

                   (iv) A Í AUB , B Í AUB

                   (v) If AÍ B and xÎ A, then x Î B

                   (vi) A Í A

                   (vii) A Ç  B Í A  ,  AÇB Í B

[10] Power set:

            The collection of all the subsets of a set A is called the power set of A. It is denoted by P(A).

Example: A = {a, b, c}

                P(A) = {f, {a}, {b}, {c}, {a,b}, {b,c}, {c, a}, {a, b, c}}

Note:   The cardinality of the power set of a set A having n elements is 2ⁿ.

[11] Universal set  (U  / S  / x / w):

A set that contains all sets in a given context is called the universal set, denoted by E or U.

Example: When we study 2-dimensional co-ordinate geometry, then the set of all the points in xy-plane is the universal set.