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BodhiAI
Nov. 11, 2019

Introduction & Def of Sets,Set Operations,Types of sets,Algebra of Sets:

Introduction:-

Suppose, you are asked to locate the odd one out of the group containing some horses and a cow in the same group, then what will be you answer. Obviously, I think, your answer will be that cow.

Now, let us assume some different situation i.e. a group containing some horses, a cow and a parrot. So, now if I ask you to please locate the odd one out, then certainly your answer will be that parrot. Now, the question arises, Why? Why not that same cow in the second situation also? Why only that bird? I think, this thing needs no explanation.

Let us extend this idea to our mathematical world now in which we refer to each and every thing as an object. The idea is "A set", A set thus is a well defined collection of object. I will fail in framing a concept if I do not introduce the phrase "well defined" in the definition.

All the objects that form a set are called its elements or members.

We usually denote sets by capital letters and their elements by small letters.

Note: If x is an element of a set A, we write x A, which means that x belongs to A or x is an element of A.

Example: a) If A is a set containing every fifth character of the English alphabets, then the set A can be represented by A = {E J O T Y}

b) If B is a set containing only one digit prime natural numbers, then the set B can be represented as B = {2 3 5 7}

c) A collection of honest people of India, is not a set, since the term honesy is vague and is not well defined. Similarly, "rich persons"; "good looking people", "good players" etc. do not form sets.

Important

In general, the most commonly used sets in mathematics are

The set of Denoted By Defined as

a) Natural Numbers N {1, 2, 3, 4, ......}

b) Whole Numbers W {0, 1, 2, 3, 4, ......}

c) Integers I or Z or Z {......, -3, -2, -1, 0, 1, 2, 3,.....}

(i) Positive Integers I+ or Z+ or Z+ {1, 2, 3, 4, .....}= N

(ii) Negative Integers I- or Z- or Z- {........, -3, -2, -1}

(iii) Non-Negative Integers I+ + {0} {0, 1, 2, 3, 4, ....} = W

(iv) Non-Positive Integers I- +{0} {........., -3, -2, -1, 0}

d) Real Numbers R () i.e. entire Number Line

(i) Rational Numbers Q Which are either terminating or if non terminating, must be recurring e.g. where p, q I and

(ii) Irrational Numbers R-Q or Qc Which are neither terminating nor recurring

e) Complex Numbers C The set of all Complex Numbers. Set Real numbers is a subset of the set of Complex Numbers.

How To Describe A Set

The way or method by which the information in the set can be explained mathematically is called a set. There are actually two ways in which a set can be described.

a) Tabulation Method or Roster Form

b) Rule Method or Set Builder Form

A. Tabulation Method/Roster Form:

In this method, a set is described by listing elements, within curly braces i.e. { }.

Example: a) If A is a set containing every fifth character of the English alphabets, then the set A can be represented by A = {E J O T Y}

b) If B is a set containing only one digit prime natural numbers, then the set B can be represented as B = {2 3 5 7}

Important

Rearrangements and/or Repetitions do not alter or change the description of the set

Example: The sets {E, J, O, T, Y}. {J T O Y E} or {E, J, O, Y, O, J} are same.

B. Rule Method/Set Builder Form

In this method, the property or properties satisfied by the elements of the set are listed. In such a case, the set is described by {x : P(x) holds} which is read as the set of all x such that P(x) holds where, P(x) is the characterising property.

Example: a) A set containing every fifth character of the English Alphabets can be expressed in set builder form as A = {x : x E, x = (5n)th character, n N}

where, E is a set of English Alphabets A to Z in order.

b) A set of all even natural numbers can be written as

B = {x : x N, x = 2n, n N}

Type of Set

A. Empty Set/ Null Set 

A set consisting of no element is called an Empty set or Null set. It is generally denoted by a greek symbol , called as phi. 

Example: a) Occurrence of a natural number in between two consecutive integers

i.e. {x : x N, 2 < x < 3} =

b) Finding the complex number from the set of Real numbers

i.e. {x : x R, x2 = -1}

B. Singleton Set

A set containing only a single element is called a singleton set.

Example: Occurrence of an odd integer between two consecutive even integers.

i.e. {x : x N, x = 2n + 1 and 4 < x < 6, n N}

C. Finite Set

A set in which the process of counting of elements surely comes to an end, is called a finite set.

Example: a) Number of integral factors of 1000

b) {x : x I, x is a factor of 1000}

c) Set of all persons living in India.

Important

Cardinal Number of a Finite Set

The total number of elekents in a finite set is called the Cardinal Number of Order of a finite set A and is usually denoted by n(A).

D. Infinite Set

A set in which the process of counting of elements does not comes to an end, is called an infinite set.

Example: Set of all points lying inside the circle

i.e. {x, y : x, y R, x2 + y2 < a2}

E. Equal Sets

Two sets A and B are sxaid to be equal, written as A = B, if every element of A is in B and vice-versa.

F. Equaivalent Sets

Two finite sets A and B are said to be equivalent, if number of elements in A equals the number of elements in B.

i.e. n(A) = n(B)