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PRE-CALCULUs

Course Code: 4431/9467

Contents

List of Figures List of Tables 1 Real Number System	iv vi 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	2
1.3 Examples of Rational and Irrational Numbers . . . . . . . . . . . . . .	3
1.4 Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . .	6
1.4.1 Basic Properties of Real Numbers . . . . . . . . . . . . . . . . .	6
1.4.2 Order Properties of Real Numbers . . . . . . . . . . . . . . . . .	7
1.4.3 Completeness Property of Real Numbers . . . . . . . . . . . . .	8
1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	10
1.6 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	12
1.6.1 Properties of Absolute Value . . . . . . . . . . . . . . . . . . . .	12
1.7 Self Assessment Questions . . . . . . . . . . . . . . . . . . . . . . . . .	12
2 Sets	15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	15
2.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	15
2.2.1 Properties of Sets . . . . . . . . . . . . . . . . . . . . . . . . . .	19
2.3 Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	23
2.3.1 Venn Diagrams for Subsets . . . . . . . . . . . . . . . . . . . . .	25
2.4 Self Assesment Questions . . . . . . . . . . . . . . . . . . . . . . . . . .	32

Functions 36
Limits 67
Continuity 84
Sequences and Series 94
Trigonometric Functions and their Applications 110
- Introduction………………………………………………………………………………………….. 110
- Objectives……………………………………………………………………………………………… 110
- Trigonometric Functions…………………………………………………………………………. 111
  - Trigonometric Functions Value…………………………………………………….. 111
  - Trigonometric Functions in Four Quadrants…………………………………… 112
- Domain and Range of Trigonometric Functions………………………………………… 114
- Trigonometric Functions Graph……………………………………………………………….. 114
- Trigonometric Functions Identities…………………………………………………………… 116
  - Reciprocal Identities……………………………………………………………………. 116
  - Pythagorean Identity…………………………………………………………………… 116
  - Sum and Difference Identities……………………………………………………… 116
  - Half Angle Identities………………………………………………………………….. 117
  - Double Angle Identities………………………………………………………………. 117
  - Triple Angle Identities………………………………………………………………… 117
  - Product Identities……………………………………………………………………….. 117
  - Sum of Identities………………………………………………………………………… 118
- Applications of Trigonometric functions…………………………………………………… 121
- Self Assessment Questions……………………………………………………………………… 125
Differentiation 127
Integration 150

List of Figures

1.1 Real number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Circle Inscribed in a Square . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Square Inscribed in a Circle . . . . . . . . . . . . . . . . . . . . . . . . 5

Venn Diagram…………………………………………………………………………………………. 24
Venn Diagram…………………………………………………………………………………………. 25
Venn Diagram…………………………………………………………………………………………. 25
Venn Diagram for Subsets………………………………………………………………………… 26
Venn Diagram…………………………………………………………………………………………. 26
Venn Diagram…………………………………………………………………………………………. 27
Venn Diagram…………………………………………………………………………………………. 27
Venn Diagram…………………………………………………………………………………………. 28
Venn Diagram…………………………………………………………………………………………. 28
Venn Diagram…………………………………………………………………………………………. 29
Venn Diagram…………………………………………………………………………………………. 29
Venn Diagram…………………………………………………………………………………………. 30
Venn Diagram…………………………………………………………………………………………. 30
Venn Diagram…………………………………………………………………………………………. 32
Venn Diagram…………………………………………………………………………………………. 33
Venn Diagram…………………………………………………………………………………………. 35
Arrow Diagram………………………………………………………………………………………… 40
Shaded Region Represents all the Members of the Relation L……………………… 41

3.3 f (x) = sin x.……………………………………………………………………………………………… 61

f (x) = cos x……………………………………………………………………………………………… 61
f (x) = tan x……………………………………………………………………………………………… 62
An Arrow Diagram…………………………………………………………………………………… 68

Intervals for Left-Hand and Right-Hand …………………………………………. 70
The Sandwich Theorem……………………………………………………………………………. 76
Continuity at a Point a…………………………………………………………………………….. 85
The Function f (x) is Continuous at x = 1…………………………………………………….. 87
Continuity at Points a, b and c………………………………………………………………….. 88
Graph of the Piece-Wise Function f (x)……………………………………………………….. 89
Right Angled Triangle……………………………………………………………………………. 111
Trigonometric Table……………………………………………………………………………….. 112
Trigonometric functions in four quadrants………………………………………………… 112
Trigonometric ratios in various quadrants…………………………………………………. 113
Domain and Range of Trigonometric Functions………………………………………… 114
Graph of sin θ and cos θ………………………………………………………………………… 115
Graph of tan θ and cot θ………………………………………………………………………… 115
Graph of sec θ and csc θ………………………………………………………………………… 115
Height and Distance………………………………………………………………………………. 122
Slope of a Tangent Line at a Point P……………………………………………………….. 129
Chain …………………………………………………………………………………………… 133
Graph of a Function sin x and its Derivative……………………………………………… 136
Area Under the Curve……………………………………………………………………………. 162
Area Between the Two Curves……………………………………………………………….. 163

List of Tables

8.1 Rules of Differentiation…………………………………………………………………………. 131

Chapter 1

Real Number System

Introduction

{ }

Numbers play a vital role in our daily life. Either we have been measuring the temper- ature of a certain place at certain times or we have been counting money or we have been collecting the data of the ages of human beings, we are actually dealing with the numbers. In the early ages, people used different symbols and notations to represent the numbers. With the passage of time, these symbols were transformed and modified in the form of a basic counting set represented by 1, 2, 3, 4, . . . which today is known as set of Natural numbers with representation N.

Some important numbers that are commonly used today are given by

N: Natural numbers are given by 1, 2, 3, 4, . . .

W: Whole numbers are given by 0, 1, 2, 3, 4, . . .

Z: Integers given by 0, ±1, ±2, ±3, . . .

Q: Rational numbers given by ¹ , 3, 3.5, 0, etc.

I: Irrational numbers given by √2, √1 , 2 + √3, etc.

Interesting fact is that, all these numbers can be found on a line which is called Real numbers line. For example some of the above mentioned numbers can be shown on the real number line as

Figure 1.1: Real number line

So we conclude that every number on this line is a real number and all real numbers are classified into two separate groups which are called rational numbers and irrational

numbers. Rational numbers are those which can be written in the form a

where a is

an integer and b is a non-zero integer. Rational numbers also include integers and some

daecimals. While the irrational numbers are those which cannot be written in the form where a is an integer and b is a non-zero integer.

Objectives

After completing this unit, students will be able to learn about

rational and irrational representations
real numbers and its basic properties
order properties of real numbers
completeness property of real numbers
absolute value and its properties

Examples of Rational and Irrational Numbers

The rational numbers are given the form of functions e.g., 1 , 4 . Similarly the following

diagram represents some rational numbers. 4 1

Terminating decimal numbers represent rational numbers e.g.,

15 3

1.5 = = ,

10 2

12 3

0.0012 =

10000

2500

· · ·

Non-terminating decimal numbers which are recurring (means which are repeating themselves) also represent rational numbers e.g., 1.¯5 = 1.5555 is a recurring decimal number. So we can change it into fraction which gives its rational representation.

We write 1.¯5 = 1.555 · · · = 1 + 0.555 · · · . Let us represent the decimal part as x

1.¯5 = 1 + x (1.1)

where

x = 0.555 . . . (1.2)

x = 0.5 + 0.05 + 0.005 + 0.0005 + . . . (1.3)

Multiplying both sides by 10:

10x = 5 + 0.5 + 0.05 + 0.005 + . . . (1.4)

10x = 5 + x (1.5)

9x = 5 (1.6)

x = (1.7)

Therefore, (1.1) takes the form

1.¯5 = 1 + 5

1.¯5 = 14

(1.8)

(1.9)

Hence, ¹⁴ is the required rational representation of the given non-terminating and

recurring decimal number.

Non-termin√ating decimal numbers which are also non-recurring represent irrational

numbers e.g., 2 = 1.4142 . . .. Since every natural number which is not a perfect

square cannot be calculated exactly so repre√sent√s no√n–terminating and non-recurring

decimals and hence irrational numbers e.g., 3, 5, 6, . . ..

Theorem 1.3.1. If a fraction ^p is not a whole number then its square ^p2 is not a whole

q q2

number either.

Proof. We suppose that the fraction ^p is in the reduced form or simplest form so that

· ·

p and q have no common factor other than 1. Then p² = p p and q² = q q also have no common factor because if m is any common factor of p² and q² then it would also

be a common factor of p and q. Therefore, ^p2 is not a whole number.

Now u√sing the contra-positive argument in the previous result we consider above

example 2 again.

Example 1.3.1. Let ^p

= √2, then ^p2

= 2. Since √2 is not a whole number but its

√

square 2 is a whole number. Therefore, it follows from the previous result that

not a fraction and hence not a rational number.

2 is

Example 1.3.2. Another famous irrational number is π (which is ratio between the circumference and diameter of a circle). If we consider a circle of area π (radius of this circle is 1) which is enclosed by a square whose area is 4 (side is 2) as shown in the figure.

Figure 1.2: Circle Inscribed in a Square

Similarly in the second diagram, a square of area 2 is enclosed by the same circle of area π.

Figure 1.3: Square Inscribed in a Circle

So,

Area of square enclosing circle = 4,

Area of a circle = π,

Area of circle enclosed by square = 2,

∴ 2 < π < 4.

Archimedes improved this inequality by considering regular polygon with 96 sides instead of square and he got the following inequality 3¹ < π < 3¹⁰ .

7 71

But now a days, scientists are using modern computer calculations to approximate the value of π and have succeeded to evaluate π upto 3 trillion decimal places.

For example, decimal expansion of π with respect to 1 decimal place, 2 decimal places, 3 decimal places, and so on is given by

3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, · · ·

π = 3.141592563589793238462643383279502884197169399375 · · ·

· · ·

Consider another real number which is not terminating and not recurring given by 1.01001000100001 . In this decimal number, number of zeros increase by 1 every time after decimal place. This is also an example of irrational number.

Properties of Real Numbers

Real numbers which we have classified on rational adn irrational numbers satisfy the following properties.

Basic properties of real numbers
Order properties of real numbers
Completeness property of real numbers

Basic Properties of Real Numbers

Real numbers satisfy the following basic properties with respect to addition and mul- tiplication. Let a, b and c be three real numbers i.e., a, b, c ∈ R. Then,

a + b = b + a.

Real numbers are commutative with respect to the basic operation of addition.

(a + b) + c = a + (b + c).

Real numbers are associative with respect to the basic opertaion of addition.

There exists an element 0 ∈ R such that 0 +a = a +0 = a for all the real numbers

a ∈ R.

— − −

∈ − ∈

For all real numbers a R, there exists a real number a R such that a + ( a) = ( a) + a = 0. This a is called negative element or additive in- verse of the real number a.

Similarly, we have four properties of real numbers according to the basic op- eration of multiplication.

· ·

a b = b a.

Real numbers are commutative with respect to the basic operation of multiplica- tion.

· · · ·

(a b) c = a (b c). Real numbers are associative with respect to the basic operations of
There exists an element 1 ∈ R such that 1 a = a · 1 = a, for all the real numbers

a ∈ R. This 1 is called unit element or multiplicative identity of real numbers.

For all the non-zero real numbers a ∈ R, there exists an element ¹ ∈ R such that

a · 1 = ¹ · a = 1. This ¹ is called reciprocal element or multiplicative inverse of

the real number a.

In the next property, we have operations of addition and multiplication together.

· · ·

a (b + c) = a b + a c, or

· · ·

(a + b) c = a c + b c.

Real numbers satisfy distributive property of multiplication over addition.

Order Properties of Real Numbers

∈

These properties tell us about the comparison of two or more real numbers, which real number is greater and which one is smaller. If we have a real number a R, then there are three possibilities which include

a is positive real number
a is zero
a is negative real number (i.e., additive inverse of a is positive real number).

But exactly one of these three possibilities will be true for any real number a ∈ R.

We can further generalize this rule by saying that if we have two real numbers a

and b then exactly one of the following statements will be true

a < b, (ii) a = b, (iii) a >

−

Example 1.4.1. If a = 5 and b = 3. Then exactly one of the following three statements will be true

(i) − 5 < 3, (ii) − 5 = 3, (iii) − 5 > 3.

Here, only first statement −5 < 3 is true.

Now, we write some more results about the order properties of real numbers.

Theorem 1.4.1. For two real numbers a, b ∈ R. If we have an inequality between them. Another real number c ∈ R can be added to both sides of that inequality without changing its order i.e., if a < b then a + c < b + c for all c ∈ R.

— − −

Example 1.4.2. If a = 5, b = 11 and c = 4. So, 5 < 11 implies 5+( 4) < 11+( 4)

i.e., 1 < 7.

∈

Theorem 1.4.2. For three real numbers a, b, c R. If we have inequalities between 1^st and 2^nd (i.e., a and b) and between 2^nd and 3^rd (i.e., b and c) then we have same inequality between 1^st and 3^rd. i.e., if a < b and b < c then we have a < c.

Example 1.4.3. If a = 2, b = 5 and c = 11. So, 2 < 5 and 5 < 11 yields 2 < 11.

Theorem 1.4.3. For two real numbers a and b. Suppose we have an inequality then we can multiply both sides if that inequality by a positive real number c without changing the order of that inequality i.e., if a < b and c > 0. Then ac < bc.

Example 1.4.4. If a = 5, b = 7 and c = 10 > 0. So, 5 < 7 and 10 > 0 implies

5 · 10 < 7 · 10, (1.10)

50 < 70. (1.11)

Theorem 1.4.4. For two real numbers a and b. Suppose we have an inequality then multiply both sides of that inequality by a negative real number c then the order of inequality is reversed. i.e., if a < b and c < 0. Then ac > bc

Example 1.4.5. If a = 5, b = 7 and c = −10 < 0. So, 5 < 7 and −10 < 0 yields

5 · (−10) < 7 · (−10), (1.12)

−50 > −70. (1.13)

Completeness Property of Real Numbers

Completeness property of real numbers means that the set of real numbers is a complete set in the sense that if we take a subset of real numbers for which we can find upper bounds or lower bounds then we can also find the smallest of these upper bounds and similarly the largest of these lower bounds within the set of real numbers. For example, some subsets of real numbers are given here:

(i) [1, 2], (1, 2), [1, 2), (1, 2]

(ii) [0, ∞), (−∞, 0]

(iii) N = {1, 2, 3, . . .}

Q = {p: p, q ∈ Z and q /= 0}

Now, in these examples we find first some upper and lower bounds and then lowest upper bound and greatest lower bound of these values exist.

For these open and closed intervals we see that real number 2 and any real number greater than 2 are upper bounds. So lowest upper bound exists and is 2. Similarly, we see that real number 0 is a lower bound and any real number less than zero is also lower Therefore, greatest lower bound is zero.

−∞

∞

For [0, ), we see that we cannot find its upper bounds but on the other hand we can find its lower bounds and also greatest lower bound. But for the set ( , 0] we can find upper bounds and lowest upper bound but we cannot find its lower bounds and consequently no greatest lower bound
For the set of natural numbers we note that 1 is a lower bound and all the real numbers less than 1 are also lower bounds. So we can find many lower bounds for the case of natural numbers but cannot find any upper bound for natural
For the set of rational numbers we see that it is not possible to find upper bounds and lower bounds. Therefore, lowest upper bound and greatest lower bound also do not exist

Now we describe some basic operations of addition and multiplication on real num- bers, one of which is rational number and the other is irrational number.

Theorem 1.4.5. The sum of a rational or irrational number is irrational.

∈ /

Proof. Let we denote the rational number by ^p , where p, q Z and q = 0 and the irrational number by r then we have to show that

+ r = s, where s is also irrational number

By contradiction we suppose that the sum s is a rational; number then we can write it in the form ^t , where t, u ∈ Z and u /= 0. Therefore,

+ r = ,

t p

r = u − q ,

r =

where p^j = tq − pu is an integer and q^j = uq is a non-zero integer. This shows that

r can be written in the form ^p′ where p^j, q^j ∈ Z and q^j /= 0 which means r is a rational

′q

number but this is not true since r is an irrational number. Hence our supposition that sum s is a rational number is wrong and so this sum must be an irrational number.

Theorem 1.4.6. The product of a (non-zero) rational and irrational number is irra- tional.

Proof. Let we denote this rational number by ^p , p, q ∈ Z and p, q /= 0 and the irrational

number by r then we show that

q × r = s, where s must be an irrational number

By contradiction we suppose that the product s is a rational number then we may

write it in the form ^t

where t, u ∈ Z and u /= 0, so

p t

q × r = u,

t p

r = u × q ,

r = , up

r = , qj

where p^j = tq is an integer and q^j = up is non-zero integer since both u and p are non-zero integers which means r is a rational number but it is not true since r is an irrational number. Therefore, our supposition that s is rational is wrong and hence the product s must be an irrational number.

Examples

Example 1.5.1. Let ¹

and √5 be rational and irrational numbers respectively then

their sum is given by

which is an irrational number.

1 + √5 =

1 + 5√5

Example 1.5.2. Let ¹

and √3 be rational and irrational numbers respectively then

their product is given by

1 √ √3

3 =

2 2

which is an irrational number.

Example 1.5.3. Let ¹

and ²

be two rational numbers then their sum is

1 2

+ = 1

3 3

But if we write ¹

and ²

as recurring decimals in the form 0.333 · · · and 0.666 · · ·

respectively then

· · ·

0.333

+ 0.666 · · ·

0.999 · · ·

−

· · ·

which means 0.¯9 = 0.999 represents 1. Since if 0.¯9 is represented by some real number x which is less than 1 then it means that 1 x is a non-negative number less than every positive number which concludes that

1 − x = 0 → x = 1.

So any decimal representation which has an infinite string of 9’s can always be replaced by a terminating decimal.

Note: The set of real numbers is a dense set. It means that all real numbers are con- nected to each other. In other words, it is evident that we cannot find two consecutive real numbers. As we know that in case of natural numbers we can find two consecutive natural numbers. Similarly, if we have two real numbers no matter how close to each other, we can find as many real numbers as we want between them.

Example 1.5.4. Let we have two real numbers ¹

numbers between them e.g., by taking their average

and ¹

then we can find many real

1 + 1

Average = ⁸ ⁹

17/72

= =

144

Also, if we write ¹

and ¹

into decimal form then we have

1 = 0.125 and 1 = 0.111 · · · = 0.¯1.

· · ·

Then we can find many decimal numbers between 0.125 and 0.¯1 = 0.111 e.g.,

0.122, 0.113, 0.114, . . . , 0.124 and so on.

Absolute Value

Absolute value is a function which when applied to real numbers gives nonnegative real numbers.

In other words, we say that | · | : R → R+ ∪ {0}, where R+ mean all positive real numbers i.e., for x ∈ R.

Properties of Absolute Value
1. |x| ≥ 0, ∀ x ∈ R and |x| = 0 if and only if x = 0.
2. |x + y| ≤ |x| + |y|, ∀ x, y ∈ R.
3. |xy| = |x||y|, ∀ x ∈ R.
4. |x − y| = |y − x|, ∀ x, y ∈ R.
Self Assessment Questions

Q 1. Represent the following real numbers on the Real line.

−

(i) 3 (ii) 1.5 (iii) √5 (iv) 3 1

√2 (vi)

√3 (vii)

√3 − 1 (viii) +∞

(ix) 1.¯9 (x) 10 9

Q 2. Identify the property of real numbers used in the following equalities. (i) 0 + 3 = 3 + 0

(ii) 5 + (−5) = 0

(iii) 5 1 + 1 = 5 1 + 5 1

(iv) −2 × 1 = 1 × −2 = 1

−2 −2

(v) −4 < 1

Q 3. Find the decimal representation for the following fractions.

157 50

Q 4. Convert the following decimals into fractions.

(i) 0.¯5

(ii) 0.123˙45˙ = 0.12345345345 · · ·

(iii) 3.14159

Q 5. Find a rational number between the rational numbers a

and c .

Q 6. Find an irrational number between √2 and √3.

Q 7. Separate the rational and irrational numbers from the following real numbers and also give your reasons.

(i) 0 (ii) 1

100

(iii) 100

(iv) √5 (v) √3 7 (vi) √8

(vii) log10 10 (viii) log2 7 (ix) 1.¯2

20 (xii) 1

Q 8. Solve the following inequalities.

(i) |x| < 2

(ii) |x − 7| < 3

(iii) |x + 1| < 1

(iv) |2x − 1| < 3

Q 9. Explore the rules for the

sum of two even numbers
product of two even numbers
sum of even and odd number
product of even and odd number
difference of two odd numbers
difference of two even numbers

Q 10. Consider the interval [1, 2] which is a subset of real numbers R. Find the upper and lower bounds for this interval. How many upper bounds and lower bounds we can fnd for this interval. Try to find the least upper bound and also the greatest lower bound for this interval from the bounds you have already found. Guess how many integres are there in this interval. Also guess how many rational numbers are there in this interval and lastly guess how many real numbers are there in this interval.

Q 11. What will be the possible values of remainder when a positive integer is divided by 4?

Chapter 2 Sets

Introduction

Set is now a basic and an established notation which is commonly used in mathematics. George Cantor, Russell and Frege are considered to be the founders of modern set theory and we have a very fascinating saying of Cantor about sets that “A set is a many that allows itself to be thought of as a one”.

Objectives

After completing this unit, students will be able to learn about

sets and different types of sets
properties of sets
Venn diagrams

Definition 2.2.1. A set is a well-defined collection of different (or distinct) items. We know from school math that several notations are used to represent a set. All these notations involve curly brackets.

Notations:

A = Set of Natural numbers 2. A = {1, 2, 3, . . .}
A = {x : x ∈ N}, N = Set of Natural numbers

{}

Definition 2.2.2. An empty set is a set which contains no object. It is represented by empty curly brackets or symbol phi, φ, is commonly used to represent an empty set.

Example 2.2.3. The set of all human beings who have weight at least 10000kg each. It is obviously an empty set. Similarly, the set of natural numbers less than zero is also an empty set.

Definition 2.2.4. Set membership is a symbol (ϵ) that is used to represent that these members belong to a certain set or do not belong to certain set. We often use small alphabets to represent members of a set and capital letters to represent sets itself.

Example 2.2.5. a ∈ A means that a is a member of the set A and it is read as “a

belongs to A”. Similarly, if a is not a member of the set A then we use notation ∈/ to

∈

represent it i.e., a / A it means that a is not a member of the set A and it read as “a does not belong to A”.

Example 2.2.6. 2 ∈ N and 1.5 ∈/ N where N is the set of natural numbers.

Definition 2.2.7. Two sets A and B are said to be equal sets if they have exactly the same members.

Example 2.2.8. A = {a, b, c} and B = {c, a, b}. Therefore, in set {a, b, c} there is no preference given to a over b or b over c etc. Hence, {a, b, c} is an unordered triplet.

Definition 2.2.9. (Cardinality of a Set) Let A and B be the two sets. If the number of members of both the set is same then A and B are of the same cardinality.

{ } { }

Example 2.2.10. A = a, b, c and B = 1, 2, 3 . So cardinality means number of members of a set. Such sets are called equivalent sets. For example, A and B are equivalent sets.

Definition 2.2.11. For two sets A and B, intersection means that all the members we take which belong to both sets A and B. We use the following notation to represent intersection:

A ∩ B = {x : x ∈ A and x ∈ B}

Definition 2.2.12. For two sets A and B, union means that all the members we consider which belong to either set A or set B or both. We use the following notation to represent union:

A ∪ B = {x : x ∈ A or x ∈ B}

Note: If a member belongs to both the sets A and B then it is written only once in the union.

Definition 2.2.13. When we consider all these basic operations on set, we must bound ourself about the limit under which we are performing these operations. This limit is represented by a larger set under consideration and is known as Universal set. So it consists of the possible members of all the sets under consideration.

Note: Sometimes a set can itself be a members of other sets. For example, the set

{a, {a}} has two members a and {a} respectively.

⊆

Definition 2.2.14. It is a basic relation between two sets. Suppose we have two sets A and B then A is called a subset of B written as A B if every member of A is also a member of the set B i.e.,

A ⊆ B ⇐⇒ ∀ x if x ∈ A then x ∈ B

⊆

This definition implies that every set is a subset of itself i.e., A A. But if we can fund at least one member in the set A which is not a member of the set B then we say that A is not a subset of B and it is written as a ¢ B i.e.,

A ¢ B ⇐⇒ ∃ atleast one x ∈ A s.t. x ∈/ B.

Example 2.2.15. Let A = {2, 4, 6, 8, 10} , B = {2, 4} and C = {8, 10, 12}. Here

B ⊆ A but C ¢ A.

⊂

Definition 2.2.16. Suppose A and B be two sets then A is called proper subset of B written as A B if every member of A is also a member of B and necessarily A and B are not equal sets. For example, in the previous example it is more appropriate to write B ⊂ A instead of B ⊆ A because A and B are not equal sets.

This subset relation property can be used to prove the equality of two sets by considering subset property from both sides. i.e., to prove A = B we need to show that A ⊆ B and B ⊆ A.

Example 2.2.17. Let A and B be two sets defined as

A = {n ∈ Z : n = 2p, where p is some integer}

B = {m ∈ Z : m = 2q − 2, where q is also some integer}

We prove that A = B.

For this first we show that A ⊆ B. Let n ∈ A the n = 2p for some integer p. Now we show that this n can also be expressed in the form 2q − 2 for some q ∈ Z i.e.,

n = 2q − 2, 2p = 2q − 2,

q = p + 1.

Hence, if we take q = p + 1 then

2p = 2p + 2 − 2 = 2(p + 1) − 2 = 2q − 2

=⇒ A ⊆ B.

Secondly, we show that B ⊆ A. Let n ∈ B then n = 2q − 2 for some integer q.

Then we show that n can be expressed as 2p for some p ∈ Z. i.e.,

n = 2p, 2q − 2 = 2p,

p = q − 1.

So, if we take p = q − 1 then

Hence, A = B.

2q − 2 = 2(q − 1) = 2p,

=⇒ B ⊆ A

−

Definition 2.2.18. The complement of a set A means all the members of the uni- versal set that are not in A. It is denoted by A^c = A^j = U A, where U is the universal set.

Example 2.2.19. U = {1, 2, 3, 4, . . .} = N and A = {x ∈ N : x = 2n for some n ∈ N}. Then A^c = U − A = {1, 3, 5, . . .}

Properties of Sets

With the help of basic properties of union, intersection and complement of sets we state some important properties of sets. Let A, B and C be any three sets. Then

∩ ⊆ ∩ ⊆

A B A and A B B, e., all the members in the intersection of two sets are off-course present in both sets A and B.

⊆ ∪ ⊆ ∪

A A B and B A B, .e., union of two sets A and B contains all the members of A and B.

If A ⊆ B and B ⊆ C then A ⊆ C.
A ∩ B = B ∩ A and A ∪ B = B ∪ A.
(A ∩ B) ∩ C = A ∩ (B ∩ C) and (A ∪ B) ∪ C = A ∪ (B ∪ C).
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∩ U = A and A ∪ U = U .
(A^c)^c = A.
A ∩ A = A and A ∪ A = A.
A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A.
A − B is called set difference and is defined as

A − B = {x : x ∈ A and x ∈/ B}

Then A − B = A ∩ B^c.

De Morgan’s Laws: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c.

Proof. We show that

(A ∪ B)^c = A^c ∩ B^c (2.1)

c c c

(A ∩ B) = A ∪ B (2.2)

This will establish that (A ∪ B)^c = A^c ∩ B^c So, for (2.1) let x ∈ (A ∪ B)^c,

=⇒ x ∈/ A ∪ B

=⇒ x ∈/ A and x ∈/ B

=⇒ x ∈ A^c and x ∈ B^c

=⇒ x ∈ A^c ∩ B^c

=⇒ (A ∪ B)^c ⊆ A^c ∩ B^c.

Similarly, for (2.2) let x ∈ A^c ∩ B^c

=⇒ x ∈ A^c and x ∈ B^c

=⇒ x ∈/ A and x ∈/ B

=⇒ x ∈/ A ∪ B

=⇒ x ∈ (A ∪ B)^c

=⇒ A^c ∩ B^c ⊆ (A ∪ B)^c .

(A ∪ B) − C = (A − C) ∪ (B − C).

Proof. Consider

L.H.S = (A ∪ B) − C

= (A ∪ B) ∩ C^c

= (A ∩ C^c) ∪ (B ∩ C^c)

= (A − C) ∪ (B − C) = R.H.S.

If A ⊆ B then B^c ⊆ A^c.

Proof. Let

x ∈ B^c =⇒ x ∈/ B

=⇒ x ∈/ A since A ⊆ B

=⇒ x ∈ A^c

=⇒ B^c ⊆ A^c.

(A ∪ B) − (C − A) = A ∪ (B − C)

Proof. Consider

L.H.S = (A ∪ B) − (C − A)

= (A ∪ B) ∩ (C − A)^c

= (A ∪ B) ∩ (C ∩ A^c)^c

= (A ∪ B) ∩ (C^c ∪ A^c)^c

= (A ∪ B) ∩ (A ∪ C^c)

= A ∪ (B ∩ C^c)

= A ∪ (B − C) .

A ∩ φ = φ and A ∪ φ = A.
A ∪ A^c = U and A ∩ A^c = φ.
φ^c = U and U^c = φ.

∩

Definition 2.2.20. Two sets A and B are called disjoint sets if A B = φ i.e., no member is common to both the sets.

Definition 2.2.21. If a set is divided into pieces such that these pieces are disjoint and also if we combine all these parts we get the orginal set back. Then such a group of sets is called a partition. For example, if we have a set X that is divided into parts/sets A₁, A₂, . . . , A_n such that

i=1

A₁ ∪ A₂ ∪ . . . ∪ A_n = X = Sn A_i

(ii) A_i ∩ A_j = φ ∀ i, j = 1, 2, . . . , n, i /= j

then A₁, A₂, . . . , A_n is called a partition of the set X.

Example 2.2.22. Let X = Z set of integers and consider

A₁ = {n ∈ X such that n > 0} = {1, 2, 3, . . .} A₂ = {n ∈ X such that n = 0} = {0}

A₃ = {n ∈ X : n < 0} = {−1, −2, −3, . . .}

Then,

Also,

A₁ ∪ A₂ ∪ A₃ = {1, 2, 3, . . .} ∪ {0} ∪ {−1, −2, −3, . . .}

= {0, ±1, ±2, ±3, . . .}

= Z

A₁ ∩ A₂ = {1, 2, 3, . . .} ∩ {0} = φ

A₁ ∩ A₃ = {1, 2, 3, . . .} ∩ {−1, −2, −3, . . .} = φ A₂ ∩ A₃ = {0} ∩ {−1, −2, −3, . . .} = φ

Therefore, A₁, A₂, A₃ represent a partition of the set of integers Z.

Note that we may have many partitions of a given set X. For example, for the above mentioned set of integers Z we may have other partitions.

A₁ = {n ∈ Z : n is even} = {0, ±2, ±4, . . .}

A₂ = {n ∈ Z : n is odd} = {±1, ±3, ±5, . . .}

and also another one

A₁ = {n ∈ Z : n = 3k for some integer k}

A₂ = {n ∈ Z : n = 3k + 1 for some integer k} A₃ = {n ∈ Z : n = 3k + 2 for some integer k}

Definition 2.2.23. Power set of a given set consists of all the subsets of the given set.

Example 2.2.24. If A = {1, 2} then it has four subsets given by φ = {}, {1}, {2}, {1, 2}. So the power set of A = {φ, {1}, {2}, {1, 2}}.

Definition 2.2.25. If a set X has n members then its power set has 2ⁿ members.

For example, in the previous example given set A has two members so its power set has 2² = 4 members as shown there.

Definition 2.2.26. (Inclusion/Exclusion Principle for Sets) Let A and B be any two sets with finite cardinality then

n(A ∪ B) = n(A) + n(B) − n(A ∩ B). (2.3)

Similarly, if we have any three sets A, B and C with finite cardinality then

n(A∪B∪C) = n(A)+n(B)+n(C)−n(A∩B)−n(A∩C)−n(B∩C)+n(A∩B∩C). (2.4)

In principle (2.3), if we know any two terms then we can find the third one. Sim- ilarly, in principle (2.4) if we know any seven terms then we can calculate the eighth term.

{ } { }

Example 2.2.27. Let A = 2, 4, 6, 8, 10, 12 and B = 4, 8, 9, 10, 11 . Here n(A) = 6,

n(B) = 5,.

Then

A ∩ B = {4, 8, 10}

n(A ∩ B) = 3

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

= 6 + 5 − 3

n(A ∪ B) = 8

i.e., A ∪ B = {2, 4, 6, 8, 10, 11, 12}

Venn Diagram

Many problems of set theory cam be solved by using the diagrams. This idea was given by the British mathematician John Venn in 1881. That is why these diagrams are called Venn Diagrams. The basic concept is to represent sets in the form of the regions in the coordinate plane. Mostly, we use rectangular region to represent the universal set and circular regions to represent the sets A, B, C etc. i.e., ractangular region contains all the circular regions into it.

and

Example 2.3.1. Consider the universal set U = {x ∈ N : 1 ≤ x ≤ 12} = {1, 2, 3, . . . , 12}

A = {2, 4, 6, 8, 10, 12}

B = {4, 8, 9, 10, 11}

∪ ∩

then we can find A B, A B, A^c, B^c and many other sets with the help of Venn diagram.

Figure 2.1: Venn Diagram

Here Venn diagram consists of four regions.

Region which is enclosed by the rectangle U excluding both the circular regions is called Region 1 and is represented by U − {A ∪ B}.
Region represented by the circular region A excluding the circular region B is called Region 2 and is represented by A − B.
Region represented by the circular region B excluding the circular region A is called Region 3 and is represented by B − A.
Region represented by common region of both circular regions A and B is called Region 4 and is given by A ∩ B.

Now from Venn diagram, we observe that

A ∪ B = {2, 4, 6, 8, 9, 10, 11, 12} = Region 2 + Region 3 + Region 4

A ∩ B = {4, 8, 10} = Region 4

A^c = {1, 3, 5, 7, 9, 11} = Region 1+ Region 3

B^c = {1, 2, 3, 5, 6, 7, 12} = Region 1 + Region 2.

Venn Diagrams for Subsets

We represent subsets with the help of Venn diagrams. Let A and B be any two sets then the subset relation A ⊆ B can be represented by Venn Diagrams

Figure 2.2: Venn Diagram

Similarly, if A ¢ B then it can also be represented by any one of the following Venn diagrams.

Figure 2.3: Venn Diagram

Example 2.3.2. Let U = Set of real numbers = R. The we can represent Rational numbers Q, Integers Z, Whole numbers W and Natural numbers N with the help of a Venn diagram as shown.

Figure 2.4: Venn Diagram for Subsets

Now we represent all the basic operations of sets which we have studied in this unit with the help of Venn diagrams. Let A and B be any two sets then

A ∪ B is represented by the shaded region in the following Venn

Figure 2.5: Venn Diagram

A ∩ B is represented by the following shaded

Figure 2.6: Venn Diagram

A − B or A ∩ B^c is shown by the following shaded

Figure 2.7: Venn Diagram

4. De Morgan’s Laws

∪ ∩

(A B)^c = A^c B^c.

Left hand side is represented by

Figure 2.8: Venn Diagram

Right hand side is represented with the shaded region.

Figure 2.9: Venn Diagram

∩ ∪

(A B)^c = A^c B^c.

Left hand side is represented by the shaded region.

Figure 2.10: Venn Diagram

Right hand side is represented by the shaded region.

Figure 2.11: Venn Diagram

∪ ∪

Let we have any three sets A, B and Cthen A B C is shown by the following shaded region:

Figure 2.12: Venn Diagram

A ∩ B ∩ C is represented by the shaded region:

Figure 2.13: Venn Diagram

Here in case of three sets, the Venn diagram is divided into eight regions. If we know the members of any seven shaded regions. Then we can easily find the members of the missing shaded region by using the inclusion/exclusion principle (2.4).

Example 2.3.3. During a survey of a college class consisting of 50 students. It has been observed that 22 students like Mathematics, 20 students like Physics, 25 students like Biology, 5 students like Mathematics and Physics, 6 students like Physics and Biology, 7 students like Mathematics and Biology and 53 students like at least one of the above mentioned three subjects. Then how many students like all three subjects. How many students like Mathematics only. How many students like Physics and Biology but not Mathematics.

Let

Then,

Total strength of class = 60,

No. of students who like no subject = 60 − 53 = 7.

M = Set of student who like Mathematics

P = Set of student who like Physics

B = Set of student who like Biology

M ∩ P = Set of students who like both Mathematics and Physics

P ∩ B = Set of students who like both Physics and Biology

M ∩ B = Set of students who like both Mathematics and Biology

M ∪ B ∪ P = Set of students who like Mathematics or Physics or Biology

M ∩ B ∩ P = Set of students who like Mathematics and Physics and Biology Now by using Inclusion/Exclusion Principle,

n(M ∪ P ∪ B) = n(M ) + n(P ) + n(B) − n(M ∩ P ) − n(P ∩ B) − n(M ∩ B)

+ n(M ∩ P ∩ B)

53 = 22 + 20 + 25 − 5 − 6 − 7 + n(M ∩ P ∩ B)

53 = 49 + n(M ∩ P ∩ B) n(M ∩ P ∩ B) = 4

Hence, there ate four students in class who like all the three subjects (Mathematics, Physics and Biology).

Now we draw a Venn diagram and fill all the regions by starting from the number

n(M ∩ P ∩ B) = 4 inside the region no. 8.

Figure 2.14: Venn Diagram

Since there are 5 students who like Mathematics and Physics so we fill the region no. 6 by 1 and so on. From the Venn diagram we see that there are 14 students who like Mathematics only. Similarly, there are 2 students who like Physics and Biology but not Mathematics.

Self Assesment Questions

Q1. Write down all the members of the following sets.

(i) A = {x ∈ Z : 0 ≤ x < 7}

(ii) B = {x ∈ Z : −1 < x ≤ 2}

C = {x ∈ Z : n = (−1)^q + 1 for some integer q}

(iv) D = {x ∈ Z − 2 < x < 4}

Q2. Check whether the following subset relations are true or false. (i) {1} ⊆ {2, 1}

(ii) {1} ⊆ {{2}, {1}}

(iii) {1} ⊆ {1}

Z ⊆ Q
Q ⊆ Z
N ⊆ Q
Q ⊆ R

Q3. Let A = {m ∈ Z : m = 2p + 1 for some integer p} and B = {n ∈ Z : n = 2q − 1 for some integer q}. Then check the following A = B.

{ ∈ } { ∈ }

Q4. Let A = n Z : n is divisible by 2 and B = m Z : m is divisible by 4 .

Then check the following relations.

A ⊆ B, (ii) A ⊆ B, (iii) A =

Q5. Consider the Venn Diagram then shade the following sets separately on the given Venn diagram.

‘

A ∪ B

Figure 2.15: Venn Diagram

(A ∩ B)^c
A − (B ∪ C)
A^c ∩ B^c ∩ C^c
B^c ∪ C^c

Q6. Prove the properties of sets (1–10) given in the section 2.2.1 by using the Venn Diagrams.

Q7. Find at least two properties for each of the following sets.

(i) E = {n ∈ Z : 0 ≤ n ≤ 1000}

R = Set of all real numbers

M = {a, b, c, d, e, f }

Q8. Find all the power sets of the following sets.

(i) A = 1 , 1

B = {x ∈ Deck of cords : x is a king cord}
C = {n ∈ Z : n = n²}

Q9. Find union and intersection of the following collection of sets.

A = x ∈ R : − 1 < x < 1 = − 1 , 1 , i = 1, 2, 3, . . ..

Q10. A study on 77 AIOU students reflected the following data: 25 students read The News, 19 students read Express Tribune newspaper, 27 students do not read Frontier Post, 11 read The News but not Express Tribune, 11 read Express Tribune and Frontier Post, 13 read The News and Frontier Post, 9 read all three newspapers. Find how many students read none of the newspapers and how many read only Frontier Post newspaper.

Q11. Find some real numbers with the help of which we can differentiate clearly the circles of N, W Z and Q in figure 2.4.

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