Anand Classes

ANAND CLASSES

# SETS

A set is well defined class or collection of objects.

Some of the Basic Examples of Set are as follows :

• The collection of negative numbers.
• The collection of people born before 1995.
• The collection of the greatest football players
• The number of oceans  in the world
• Number of students in a class
• Collection of English vowels

All of the above collections are sets. However, the collection of the greatest football players is not well-defined. Usually, we restrict our focus to just well-defined sets.

A set is often described in the following three ways.

(i)  Descriptive Form

Example: The set of all odd counting numbers between 2 and 12.

(ii) Roster method or Listing method : In this method a set is described by listing elements, separated by commas, within braces {}. The set of vowels of English alphabet may be described as {a, e, i, o, u}.

(iii) Set-builder method or Rule method : In this method, a set is described by a characterizing property P(x) of its elements x. In such a case the set is described by {x : P(x) holds} or {x | P(x) holds}, which is read as ‘the set of all x such that P(x) holds’. The symbol ‘|’ or ‘:’ is read as ‘such that’.

The set  can be written as .

Example: If set S has all the elements which are even prime numbers, it is represented as:

S={ x : x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

# Set Theory Mathematical Symbols

There are several symbols that are adopted for common sets. They are given in the table below:

Table 1: Symbols denoting common sets

 Symbol Corresponding Set N Represents the set of all Natural numbers i.e. all the positive integers.This can also be represented by Z+.Examples: 9, 13, 906, 607, etc. Z Represents the set of all integersThe symbol is derived from the German word Zahl, which means number.Positive and negative integers are denoted by Z+ and Z– respectively.Examples: -12, 0, 23045, etc. Q Represents the set of Rational numbersThe symbol is derived from the word Quotient. It is defined as the quotient of two integers (with non-zero denominator)Positive and negative rational numbers are denoted by Q+ and Q– respectively.Examples: 13/9. -6/7, 14/3, etc. R Represents the Real numbers i.e. all the numbers located on the number line.Positive and negative real numbers are denoted by R+ and R– respectively.Examples: 4.3, π, 4√ 3, etc. C Represents the set of Complex numbers.Examples: 4 + 3i, i, etc.

# Types of sets

(1) Null set or Empty set : The set which contains no element at all is called the null set. This set is sometimes also called the ‘empty set’ or the ‘void set’. It is denoted by the symbol  or { } or Ø.

(2) Singleton set : A set consisting of a single element is called a singleton set. The set {5} is a singleton set.

(3) Finite set : A set is called a finite set if it is either void set or its elements can be listed (counted, labelled) by natural number 1, 2, 3, … and the process of listing terminates at a certain natural number n (say).

Cardinal number of a finite set : The number n in the above definition is called the cardinal number or order of a finite set A and is denoted by n(A) or O(A).

(4) Infinite set : A set whose elements cannot be listed by the natural numbers 1, 2, 3, …., n, for any natural number n is called an infinite set.

(5) Equivalent set : Two finite sets A and B are equivalent if their cardinal numbers are same i.e. n(A) = n(B).

(6)   Equal set : Two sets A and B are said to be equal iff every element of A is an element of B and also every element of B is an element of A. Symbolically, A = B.

(7) Universal set : A set that contains all sets in a given context is called the universal set.

It should be noted that universal set is not unique. It may differ in problem to problem.

(8) Power set : If S is any set, then the family of all the subsets of S is called the power set of S.

The power set of S is denoted by P(S). Obviously Ø and S are both elements of P(S).

Example :  Let S = {a, b, c}, then P(S) = {,Ø {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.

Power set of a given set is always non-empty.

(9) Subsets (Set inclusion) : Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B.

A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part of another set.

Example: A = {1,2,3}

Then {1,2} ⊆ A.

Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.

Note: The set is also a subset of itself.

If A is not a subset of B, then it is denoted as A⊄B.

(10) Proper Subset : If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.

Example: If A = {2,5,7} is  a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}

But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.

(11) Superset : Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B.

For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.

(12) Universal Set : A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values.

Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:

U = {1,2,3,4,5}

# Operations on Sets

In set theory, the operations of the sets are carried when two or more sets combine to form a single set under some of the given conditions. The basic operations on sets are:

• Union of sets
• Intersection of sets
• A complement of a set
• Cartesian product of sets.
• Set difference

## Union of Sets

If set A and set B are two sets, then A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:

A ∪ B = {1,2,3,4,5,6}

## Intersection of Sets

If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B.

Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:

A ∩ B = { } or Ø

Since A and B do not have any elements in common, so their intersection will give null set.

## Complement of Sets

The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’.

Properties of Complement sets

1. P ∪ P′ = U
2. P ∩ P′ = Φ
3. Law of double complement : (P′ )′ = P
4. Laws of empty/null set(Φ) and universal set(U),  Φ′ = U and U′ = Φ.

## Cartesian Product of sets

If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B.

We can represent it in set-builder form, such as:

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: set A = {1,2,3} and set B = {Red, Yellow}, then;

A × B = {(1,Red),(1,Yellow),(2,Red),(2,Yellow),(3,Red),(3,Yellow)}

## Difference of Sets

If set A and set B are two sets, then set A difference set B is a set which has elements of A but no elements of B. It is denoted as A – B.

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

A – B = {1,5}

# Set Theory Notations

 Symbol Symbol Name { } set A ∪ B A union B A ∩ B A intersection B A ⊆ B A is subset of B A ⊄ B A is not subset B A ⊂ B proper subset / strict subset A ⊃ B proper superset / strict superset A ⊇ B superset A ⊅ B not superset Ø empty set P (C) power set A = B Equal set A′,  Ac Complement of A a∈B a element of B x∉A x not element of A

# Set Theory Formulas

Some of the most important set formulas are:

 For any three sets A, B and C n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B) If A ∩ B = ∅, then n ( A ∪ B ) = n(A) + n(B) n( A – B) + n( A ∩ B ) = n(A) n( B – A) + n( A ∩ B ) = n(B) n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B ) n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

Problems and Solutions

Q.1: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).

Solution: A ∩ B = {a, b, c} ∩ {c, d, e, f}

A ∩ B = { c }

A ∩ C = { a, b, c } ∩ { c, d, e }

A ∩ C = { c }

∴ (A ∩ B) ∪ (A ∩ C) = { c }

Q.2: Give examples of finite sets.

Solution: The examples of finite sets are:

Set of months in a year

Set of days in a week

Set of natural numbers less than 20

Set of integers greater than -2 and less than 3

Q.3: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then find (A – B)′.

Solution: A – B is a set of member which belong to A but do not belong to B

∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}

A – B = {3, 5}

According to formula,

(A − B)′ = U – (A – B)

∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}

(A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

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