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### Number SystemChapter 1

9th Grade - Mathematics

Unit 1 –Square Roots

- Objectives
- Identify square numbers
- Find the square root of a perfect square number by factor method
- Find the square root of perfect square numbers, decimals and numbers which are not perfect squares by division method
- Find the smallest number that has to be added to or subtracted from a given number to make it a perfect square number
- Explain the process of finding the square root by division method

Recalling

- Squares and Square roots
- If a number is multiplied by itself, the product so obtained is called the square of that number, or a perfect square number is the product of two equal factors.

Example : 2 x 2 = 4 (—2) x (—2) = 4

- The square root of a given positive real number will have two values : positive and negativewhereas the symbol is used to indicate positive square root only.

Example is +2 or—2

Recalling

- Squares and Square roots
- If a perfect square number has ‘N’ digits then its square root has

digits when ‘N’ is even digits when ‘N’ is odd

- We can find the square root of a number by using the factor method

Squares & Square roots - Exercises

- Find the squares for the following numbers

15 0.6 0.22 0.0082 5 / 14 102/61

- Express the following in n2form

25 10,00,000

- Find the square root using factor method

254842704980138025211600

- Find the value of the following

+-

x

Square root using Division Method

- Recall Factor Method
- Procedure
- First find prime factors
- Write number as product of prime factors
- Pair equal primes
- Ex: 47,43,68,400 = 2x2 x 2x2 x 3x3 x 3x3 x 5x5 x 11x11 x 11x11
- Write square root
- Ex: = 2x2x3x3x5x11x11 = 21780
- Limitations
- Lengthy
- Not convenient for very large numbers
- Unsuitable for numbers that are not perfect squares
- Ex: 5 8 10 11

What is Division Method ?

- Starting from the unit place and moving to the left, group two digits at a time.
- Place a bar over pairs of digits, which forms a group and the left over single digit forms another group.
- Begin the division process from the highest place value. In 361, 3 is in the highest place.
- The largest square less than or equal to the first group i.e., 3, it is 1.
- Take the square root of 1, i.e., 1 as the divisor and quotient, write this quotient 1 above the first group 3

… What is Division Method ?

- Find the product of the divisor and quotient, i.e. 1 x 1 = 1
- Subtract this product from the first group, i.e., 3. The remainder is 2
- Bring down the next group 61 to the right of remainder 2.
- Double the quotient 1. It is 1+1 = 2
- Enter it in the divisor column with a blank space on the right for the next digit

… What is Division Method ?

- Find the next digit in the quotient. Let the digit be ‘a’.
- Find ‘a’ such that 2a x a = 261
- Since 261 has a ‘1’ in its units place ‘a’ should be either 1 or 9
- Check: 21 x 1 = 21

29 x 9 = 261

- Hence a = 9
- Therefore, Write 9 in both the divisor and quotient

… What is Division Method ?

- Multiply the new digit 9 in the quotient with the second divisor 29
- 9 x 29 = 261
- Now subtract 261 from 261 giving remainder ‘0’
- Therefore, the division process is completed.
- The quotient obtained by this division process is 19.
- This is the square root of 361
- Therefore, = 19

… What is Division Method ?

- Using the following table guess the digit in the units place of the square root of a given square number

Division Method - Examples

Example :

Note:

If the number is large, then continue the steps 3,4,5 and 6 till the last group has been dealt with. The available quotient will be the square root of the given number,

In this method, the groups have been made by placing the bars over each pair of digits.

The number of bars (or groups) will be equal to the number of digits in the square root.

Division Method - Examples

Example :

Note:

Note : To find the third divisor, double the digit in units place of the second divisor,

i.e., 45=40+5

Therefore 40 + 5 x 2 = 50

Or Simply add 5 to 45

i.e., 45 + 5 = 50

then find the next digit i.e.. 8

Division Method - Examples

Example :

Division Method - Examples

Example : Find Side of a square with area 55,225 square metres

Area of Square= side x side = side2

side2 = 55225

side =

side = 235 mts

Division Method – Application

- Find the least number that should be subtracted from a number to make it a perfect square, For Example – 2361

- Using Division Method the remainder is 57
- So 57 is the least number that should be subtracted from 2361 to make it a perfect square 2304 whose square root is 48

Division Method – Application

- Find the least number that should be added to a number to make it a perfect square, For Example – 7390

- Using Division Method the remainder is 165 and quotient 85
- Since remainder is great than 0 the number is between 852 and 862.
- So the least number is 862 – 7390 = 6
- So 6 is the least number that should be added to 7390 to make it a perfect square 7396 whose square root is 86

Division Method - Exercises

- Find the number of digits in the square root of these perfect squares

225 2304 6924 17424 44100 75625

- Square root using division method

441 3844

99856 219024

- Least number to subtract from these to make it perfect square

27092815423550847401

- Least number to be added to these to make it perfect square

11513479636082809215

Exercises - Application

- Length of square whose are is 41,209 SqMts
- Commander wants army of 15,376 soldiers to assemble in square form. How many soldiers in each row ?
- 13456 students are sitting in a stadium where the number of students per row is same as number of rows. How many rows ?
- Gardener arranges plants in rows to form a square. Doing this 5 plants are left out. If total plants are 64014 then find number of plants in each row ?

Square root of Decimal Numbers

- Some decimal numbers are their squares are given. Observe the number of digits after the decimal point in the square. What do you observe ?
- We can conclude that number of digits in the decimal part of a decimal square is twice the number of digits in the decimal part of the decimal number.

Arriving at square root of decimal numbers

- Take an example
- Ignoring decimal point we get is 144
- Find which is 12
- Place the decimal point using the property we saw in the above rules i.e. = 1.2
- Another example
- Ignoring decimal point we get is 36
- Find which is 6
- The square root should have 4 digits in the decimal parti.e. = 0.0006

Square root of decimals by division method

- First step is to group the digits into given number of pairsEx: 252.70729
- For whole numbers we move from units digits to the lefti.e. 252 2 52
- In case of decimal we move from decimal to the righti.e. .70729 7 07 29
- After grouping the process of finding the square root is the same as for whole numbers

Square root of decimals by division method

- Grouping the digits in pairs
- The square root of 81 i.e., the first group is 9. Write 9 in both divisor and quotient columns. Now multiply these 9s and enter the product 81 below the first group.

Square root of decimals by division method

- Subtract 81 from 81, giving a remainder 0.Now bring down the next group 54.Since the integral part is exhausted and 54 is the decimal part of the number, place a decimal point in the quotient.i.e. 9 .

Square root of decimals by division method

- Double the quotient 9 and enter it in the divisor column. Since we have to write a new digit in the divisor column and by doing so, the divisor will be more than the dividend 54, we bring down the next group 09 also. So place one zero in the quotient column and one zero in the divisor column. Bring down the next group 09.

Square root of decimals by division method

- Let the new digit in the quotient be ‘a’find a such that 180ax a = 5409x is either 3 or 9check: 1803 x 3 = 5409Write 3 in both divisor and quotient columns.Subtracting 5409 we get ‘0’.Quotient: 9.03 which is

- Examples

Example :

Note:

There are two zeros after decimal point.

Hence write 0 after decimal point of the quotient and continue the steps of finding square root by division method.

- Exercises

- Determine the number of digits in the decimal part of the square root:

84.8241 0.008281 1227.8016 144.0976312516

- Find the square root of the following by division method:73.96235.31560.000017640.00002116

Application of concept

- Find length of one side of square play ground with area 150.0625 square metres
- A number is multiplied by itself giving a product 47.0596
- Find the perimeter of a square garden whose area is 1227.8016 square metres

Finding square root of non-perfect squares

Find the square root of 2

- Think of the largest square less than or equal to 2. It is 1.
- Take its square root. = 1
- Write 1 as divisor and also as quotient.
- The product of the divisor and quotient is 1
- Write 1 below 2 and subtract
- The remainder is 1

Finding square root of non-perfect squares

- Double the quotient 1. It will be 2. Write 2 in the divisor column

(Note: The divisor 2 is bigger than the remainder 1. In this situation we cannot proceed further as the remainder is smaller than the divisor. In order to proceed further to find the square root of 2, we have to write a pair of zeroes after placing a decimal point)

- Place a decimal point after 1 in the quotient. Write two zeroes after placing a decimal point next to the dividend 2.Bring down these two zeros (a group)

Finding square root of non-perfect squares

- Let the new digit in the quotient and in the divisor column be ‘a’Find ‘a’ such that 2a x a ≤ 100By Inspection, we will find that 24 x 4 = 96 < 100
- Subtract 96 from 100. The remainder is 4. Double the quotient without considering the decimal point.

Finding square root of non-perfect squares

Note: Again we have arrived at a situation, where we cannot proceed further as the remainder is smaller than the divisor. To proceed further, write two more zeroes after 2.

- Bring down these two zeroes (i.e., a group). Let the new digit in the quotient and in the divisor column be ‘a’. Now find ‘a’ such that28a x a ≤ 400
- The new digit in the quotient column and in the divisor column will be 1. Write the product of 281 and 1 below 400.

Finding square root of non-perfect squares

- Subtract 281 from 400. Double the quotient without considering the decimal point and write it in the divisor column.
- As the remainder is smaller than the divisor, write two more zeroes, after 2.0000. Bring down these two zeroes. (i.e., a group)

Finding square root of non-perfect squares

- Find a such that, 282a x a <= 11900 By inspection we find that a = 4write this 4 both in the quotientand in the divisor columns.
- Subtract the product of 2824and 4 from 11900

Note: Since is an irrational number, the division process will be non-terminating and non-recurring. Hence we continue the division process for required number of decimal places.

i.e., If the square root value is required for ‘n’ decimal places we continue the division process up to (n+ 1) decimal places and then correct it to ‘n’ places. In the above example the quotient found is 1.414 which has 3 decimal places. We can correct it to 2 decimal places as 1.41.

= 1.41 (correct to 2 decimal places)

- Examples

Example : corrected to 2 places

Example : corrected to 2 places

Note : We have to continue division up to 3 decimal places. Already there is one pair of digits after decimal point. Therefore we take two more pairs of zeroes.

- = 2.236 up to 3 decimal places
- = 2.24 corrected to 2 places

- Exercises

- Find the square root of the following corrected to 2 decimal places:

3 6 7 10 11 20 24 25.36 0.8 0.0041

Application of concept

- Find approximate length (correct to 1 decimal place) of square plot whose area is 325 sq mts
- Find the approximate length of the side of a square whose area is 12.0068 sq mts (correct to 2 decimal places)

Unit 2 – Real Numbers

- Objectives
- Recall different sets of numbers
- Differentiate the numbers as rational and irrational
- Realisethe need of a new set of numbers called set of real numbers
- Explain the meaning of real numbers
- Explain the properties of real numbers

Recalling Different Sets of Numbers

- W is the set of Whole Numbers written asW = {0, 1, 2, 3, 4, 5 . . . . . }
- The set of whole numbers is not closed under operation subtraction.For example 2-10 = -8
- This problem can be understood by defining the set of Integers, which is representedby Z.Z = { . . . . -4, -3, -2, -1, 0, I, 2, 3, 4, . . . .)Now, we have 2 – 10 = - 8 Zwe also know that W Z
- A further extension of set of integers is needed, because many division problems withintegers do not have answer in Z.

Rational Numbers

- = 2 is not an integer. To include such numbers consider the set of rational numbers defined as follows:
- A Number is called rational number if it can be written in the form where a,b Z and b 0 (also a and b have no common factor other than 1)
- Examples: = 0.1 = 0.5 = 3.33… etc
- Therefore a rational number is either a terminating decimal or a non-terminating recurring decimal
- Now: W Z Q

Irrational Numbers

- There are some numbers such as ,,, , etc. which cannot be expressed by the form a/b (where a and b are integers and b 0)
- Such numbers are called irrational numbers
- A Number is called irrational number if it has non-terminating and non-recurring decimals
- Examples: = 1.414213…… = 2.64575…… = 3.1416……

Real Numbers

- The two sets of numbers discussed earlier, Rational and Irrational numbers are grouped into one set called ‘Real Numbers’
- Therefore the set of real numbers ‘R’ contains all terminating decimals, non-terminating recurring decimals and non-terminating non-recurring decimals
- This entire set of numbers represented by decimals is called the set of real numbers
- To every point on the number line there is a corresponding real number and vice-versa
- Thus the collection of all points on the number line can be thought of as the system of real numbers

There is nonumber, that is bothrational and irrational.

Relationship between numbers – Venn Diagram

The relationship between the set of real numbers and its subsets can be represented using Venn diagrams as shown below.

Relationship between numbers – Flow Diagram

The relationship between the set of real numbers and its subsets can be represented using Venn diagrams as shown below.

Properties of real numbers

- Properties of Equality
- If a = a, Reflexive property
- If a = b and b = a, Symmetric property
- If a = b and b = c then a = c, Transitive property
- If a = b then a+c = b+c and ac = bc
- If ac = bc and c 0, then a = b

Real Numbers - Exercises

- State the basic property of R used in each of the following statements:

Real Numbers - Exercises

- a. Give the additive inverse of each of the following:b. Give the multiplicative inverse of each of the following:
- Does the commutative property hold good on the set R for the following operations. a) Subtraction b) Division. Justify your answer.
- Is addition distributive over multiplication? Justify your answer.

Real Numbers - Exercises

- Say true or false:

Real Numbers - Exercises

- Verify the indicated property:

Unit 3 – Surds

- Objectives
- Define a surd
- Explain the meaning of surds.
- Identify the order and radicand of a surd
- Express the surds in the index form and vice versa
- Differentiate between pure and mixed surds
- Express a pure surd in the mixed form and vice versa
- Differentiate between like and unlike surds
- Locate irrational numbers,, , etc. on the number line.

Recalling Rational and Irrational Numbers

- You know that the set of real numbers consists of rational numbers and irrational numbers
- Numbers like,, , etc.are called rational numbers.
- Numbers like , , , , etc.are called irrational numbers.
- Interesting to note that there is no rational number whose square is 2. But there is a real number written in form 1.4142135… whose square is 2.

Are these similar types ?

- Now observe the two types of numbers given in Column A and Column B. What type of numbers are they ?

- Both the columns appear to have the same type of numbers. i.e., a rational number with a root sign.
- But carefully observe the numbers in column A and find their values.
- Therefore these are rational numbers

… Are these similar types ?

- Now let us consider the numbers in column B. These numbers do not have definite values because they are irrational numbers. i.e., they are non-terminating and non-recurring decimals.
- We say that, this special category of numbers have rational numbers with irrational roots, such numbers are called surds.

A Surd is defined as the

irrational root of a

rational number

Are these different ?

- In the table given below columns A and B contain irrational numbers.Then what is the difference ?

- Here, column A contains surds, as they are irrational roots of rational numbers. Column B contains irrational roots of irrational numbers. Hence they are not surds.
- The general form of surd is Whereroot sign n order a radicand[Hence a > 0 and n > 1]

Every surd is an irrational number, but every irrational number is notnecessarily a surd.

For Study …

- In the following table of surds, their order and radicand of the respective surds arc given. Study them.

Surds - Exercises

- Convert the following surds into index form:
- Express the following in the surd form:
- Express as pure surd:

Surds - Exercises

- Express as a mixed surd in its simplest form:
- Identify and group the like surds:

Index of a Surd

- Consider the product = = From the I Law of Indices= =
- Also x = =
- Therefore we can equate = x
- Or we can write =
- Similarly =

… Index of a Surd

- Ex:x x = [From the Law I of indices] = =
- Also x x = =
- Thus we can write x x = x x
- Or we can write =
- Similarly =

For Study …

- The above examples show that we can express a surd in the Index form. The index form of some surds are given below, study them.

Think! and differ. why?

Reduction of Surds

- Consider the surd We can simplify it as follows:

= 75 can be written as the product of a perfect square number and a number which is not a perfect square.

= Here 25 and 3 are two such factors 25 can be written as the square of 5

= Taking the square root of 25

Reduction of Surds - Examples

- In the above examples, identify the coefficients of the surds and the coefficients of the simplest form.
- All the surds have unity as their rational coefficient.
- Simplest form of surds have a rational coefficient other than unity.
- Some examples of pure surds are, , , ,

Surds which have unity as their rational coefficients are called pure surds.

… Reduction of Surds - Examples

- In the same table we can see how a pure surd can be converted into a mixed form.
- The reverse process helps you to convert a given mixed surd into the pure form.
- Some examples of pure surds are, , , ,

The surds which have rational coefficients other than unity are calledmixed surds.

… Reduction of Surds - Examples

- Mixed Surd to Pure Surd
- The steps to be followed to convert a given mixed surd into a pure surd are listed below.
- Take the rational factor inside the root sign.
- When it is taken inside, it is written by raising it to the index equal to the order of the given surd.
- Simplify the terms inside the root sign.

Like Surds

- Consider the surds , , ,
- Compare their radicands and their order. All these surds have the same order i.e., 2 but their radicands are different.
- Let us reduce these surds to simplest form.
- We get , , and respectively.

... Like Surds

- We get , , and respectively.
- Now compare their radicands and their order.
- What do you observe?
- The order of these surds is same i.e. 2. Similarly their radicand is also same, i.e., 3.
- Such surds are called like surds.
- Some more examples of Like Surds

Surds, whose order and radicand are same in their simplest form are called like surd

Unlike Surds

Now consider the following table. Three sets of surds are given in columns I, II and III . Observe their order and radicand.

Some more examples are:

Surds which do not have the same order and same radicand in their simplest form are called unIike surds.

Irrational Numbers on the Number Line

- We know that is an irrational number
- And is between 1 and 2

1 x 1 = 1 (Less than 2)

x = = 2

2 x 2 = 4 (Greater than 2)

- So we write 1 < < 2

… Irrational Numbers on the Number Line

- And is between 1.4 and 1.5

1.4 x 1.4 = 1.96 (Less than 2)

x = = 2

1.5 x 1.5 = 2.25 (Greater than 2)

- So we write 1.4 < < 1.5

… Irrational Numbers on the Number Line

- And is between 1.41 and 1.42

1.41 x 1.41 = 1.9881 (Less than 2)

x = = 2

1.42 x 1.42 = 2.0164 (Greater than 2)

- So we write 1.41 < < 1.42

… Irrational Numbers on the Number Line

- Moreover we know that = 1.414....... , is a non- recurring and non-terminating number. Then how do we locate this number accurately on a number line?
- If we proceed in this manner we may not locate exactly or any other irrational number on a number line.
- So to locate the actual position of irrational number on a number line, we have to think of some other method.

Using Pythagoras theorem

- Using Pythagoras theorem we can locate accurately the position of any irrational number on a number line.
- According to Pythagoras theorem, the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides.
- Therefore in the given figure,

AC2=AB2+BC2 1

Suppose, AB = BC = 1 unit

Then, in equation (1) we have

AC2= 12+12

AC2=2 or AC=

- Based on this we can locate irrational numbers on the number line

Using Pythagoras theorem - Example

- Determine on the number line, the point which represents the irrational number .

Using Pythagoras theorem - Example

- Determine on the number line, the point which represents the irrational number .

Using Pythagoras theorem - Example

Know this!

Theodorus of Cyrene, who lived around 425 B.C., was a philosopher of ancient Greece. It is said that he discovered the construction shown below which is called “The Wheel of Theodorus”. The wheel of Theodorus can be used to locate the irrational numbers on the number line as shown in the figure given below.

Using Pythagoras theorem - Exercise

Activity !!!

Construct the wheel of Theodorus in your note books and locate the possible irrational numbers on the number line.

Locate the following irrational numbers on the number line.

,, , ,

Unit 4 – Sets

- Objectives
- Define union, intersection, complement and difference of sets
- Find the union, intersection, complement and difference of sets
- Represent the union, intersection, complement and difference of sets using Venn diagram

Recalling Sets

- A set is a collection of well defined distinct objects.
- Sets are represented by
- Roster method e.g. A= {12, 14, 16, 18}
- Rule method or set builder methode.g. A = {x | x is an even number and 10 < x< 20}

… Recalling Sets

- Types of sets :
- A set, of which all the other sets under discussion are sub-sets, is called the universal set. It is represented by U.
- The sets formed by taking elements from the universal set are called sub-sets of U.

e.g.: U =. {x / x is a natural number)

A = {x / x is an even number}

Therefore A is a sub-set of U or (A U)

… Recalling Sets

- Types of sets :
- A set which has countable number of elements is called finite set.

e.g.: P = {2, 3, 5, 7, 11, 13}

- A set in which the number of elements are not countable is called an infinite set.

e.g. Q = {x / x is a rational number)

- A set which has no elements is called a null set or empty set.

e.g.: X = {set of whole numbers less than zero}

Venn Diagrams

- We have studied the different methods of representing the sets. Now let us learn how sets can be represented diagrammatically.
- Most of the ideas about sets and their properties can be represented by means of diagrams called “Venn Diagrams”.

Venn diagrams are named after the English Logician, John Venn (1834-1883).

… Venn Diagrams

- Usually, in Venn Diagrams the universal set is represented by a rectangle and the sub-sets are represented by circles, oval shapes etc.

Example: U { 1, 2, 3, 4, 5, 6, 7, 8)

A = {1, 2, 4, 8}

then the relation between U and A can be represented by Venn diagram as follows:

Where A U

… Venn Diagrams - Examples

Example 2:

U = {a, b, c, d, e, f, g, h}

A = {a, b} and B={c, e, g, h}

Note that the sets A and B are sub-sets of U and they have no common elements. Such sets we called disjoint sets.

… Venn Diagrams - Examples

Example 3:

U= {2,4,6,8,1012,14,16}

P= {2,4,6, 8} and Q= {4, 8, 12, 16}Set P and Q have common elements. i.e.. 4,8.

Hence P and Q are non-disjoint sets.

Set Operations

- Union of Sets

Consider the cricket team and volleyball team of your school. Some of the members may be common to both the learns. Suppose both the team members have to be combined fora joint photo session, who will be the members of the new group?

It is obvious that the new group will have members who play both cricket and volleyball and members who play only cricket or only volleyball.

This new group is the union of two teams.

Set Operations - Union

- Example

Consider the sets

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

By combining the elements of both the sets we get,

{1, 2, 3, 4, 5, 6, 7}

This new set is called Union of sets A and B. It is symbolically written as A B, and read as A Union B.

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

Note: The common elements 3 and 4 are written only once

Set Operations - Union

- Study Examples
- From the examples we can define the union of two sets as follows

The Union of two sets A and B is the set C which consists of all those elements which are either in A or in B or in both the sets A and B.

Set Operations - Union

- Symbolically written as

A B = { x / x A or x B both x A and B}

In the definition of union, the use of ‘or’ implies one of the following:

- x is a member of A
- x is a member of B
- x is a member of both A and B

Set Operations – Union – Special Cases

- A A = A
- A = A

Set Operations – Union – Special Cases

- If B A, then A B = A

Set Operations – Union – Venn Diagrams

Example 1:

- U={a, b, c, d, e, f, g}
- A={a, b} and
- B={d, g}

A B={a, b, d, g}

The shaded portion represents the union of sets A and B.

Set Operations – Union – Venn Diagrams

Example 2:

- U = {2, 4, 6, 8, 10, 12, 14, 16}
- A={2, 4, 6, 8} and
- B = {4, 8, 12, 16}
- A B = {2,4, 6, 8, 12, 16}
- In the diagram, the shaded portion represents A B

Set Operations

- Intersection of Sets

Consider the cricket team and volleyball team of your school. Some of the members may be common to both the learns. If we were to pick a group of players playing both games.

It is obvious that the new group will only have members who play both cricket and volleyball.

This new group is the intersection of two teams.

Set Operations - Intersection

- Example

Consider the sets

A = {1, 2, 3, 4} B = {2, 4, 6, 8}

Which are the common elements in both the sets.

Here 2 and 4 are present in both the sets.

Therefore the set {2, 4} is the intersection of sets A and B.

It is written as A B, and read as, A intersection B.

A B = { 2, 4 }

Set Operations - Intersection

- Study Examples

In the third example, C and D have no common elements. Their intersection C D has no elements.

C D is an empty set. Hence C D =

- Symbolically written as

A U B = { x / x A and x B}

The intersection of A with B is the set of all elements common to both A and B

Set Operations – Intersection – Special Cases

- A A = A
- If B A, then A B = B

Form two setsA and B such thatA B = A B

Set Operations – Intersection – Venn Diagrams

Draw a Venn Diagram to represent

P Q if P = {a, b, c, d } and Q = {c, e, g, i}

Set Operations – Multiple Sets

We can extend the operations union and intersection on more than two sets. Study the following examples.

Union and Intersections - Exercises

- Solve the following problems and draw Venn Diagrams for each of them:
- Find A U B, if A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8, 10}
- If X = {0, 25, 50, 75, 100} and Y = {0, 50, 100, 150}find X U Y.
- Given P = {Letters in the word STUDENT} and Q = {Letters in the word STUDY}; find P U Q
- If A = {3, 6, 9, 12,15} B = {2, 4, 6, 8, 10,12} find A U B
- Let M be the set of letters in the word MATHEMATICS and N is the set of letters in the word SCIENCE what is MN.

Union and Intersections - Exercises

- If S = {All Prime Numbers less than 10}T = {All Natural Numbers less than 10} find S T
- Given A = {a,b,c,d,e}, B={a,c,e,g} and C={p,q,r,s} find

1. A U B 2. A U C 3. A B 4. A C

5. B U C 6. B C

7. (A U B) U C 8. (A B) C

- If H = {2, 4, 6, 8, 10} and K = { }find i) H U K ii) H K
- If A is any set, show that A =

Union and Intersections - Exercises

- Let A = {1, 5, 7, 9}, B = {1, 2, 3, 4, 5}and C = {5, 6, 7, 8, 9, 10} then findi) A U (B U C) ii) A (B C)
- A = {x / x is a multiple of 5 and x ≤ 20} andB = {1, 3, 7, 10, 12, 15, 18, 25} findi) A U B ii) A B

Set Operations

- Complement of Sets

You know that, universal set is a set, of which all the other sets under discussion are subsets. Universal set is a basic set. Based on this set, we form many number of subsets.

For Example,

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}

Here 1, 3, 5, 7, 9 are the

elements in U which do not

belong to A.

This set {1, 3, 5, 7, 9} is called

the Complement set of A with

respect to U and denoted by A’

Set Operations – Complement – Examples

Example 2 : U = {10, 15, 20, 25, 30, 35, 40, 45, 50} and

A={10, 20, 30, 40, 50} then A’= {15,25,35,45}

From the above examples, it is clear that, if A is a subset of U, then its complement A’ is also a subset of U.

Thus, if U is the universal set and A is a subset of U, then the complement of A with respect to U denoted by A’ is defined as the set of all elements of U which are not in A.

A’ = {x/x U and x A}

Set Operations – Complement – Examples

For the above examples let us find AA’ and AA’

Example 1:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}

So A’ = {1,3,5,7,9} and So AA’ = {1,2,3,4,5,6,7,8,9,10}

We Know U={1,2,3,4,5,6,7,8,9,10}

AA’ = U and AA’=

Set Operations – Complement – Examples

Example 2:

U = {10, 15, 20, 25, 30, 35, 40, 45, 50} &

A={10, 20, 30, 40, 50} So A’= {15,25,35,45}

AA’={10,15,20,25,30,35,40,45,50}

We know U = {10, 15, 20, 25, 30, 35, 40, 45, 50}

AA’ = U and AA’=

Set Operations – Complement

A A’ = U

A A’ =

The union of a set A and its complement A’ is the Universal set U.

The intersection of a set A and its complement A’ is always a null set.

Set Operations – Complement – Examples

Now let us find the complement of A’ ,i.e. (A’)’

In the example 2 seen before

We have A’ = {15,25,35,45}

Hence (A’)’ = {x/xU and xA’}

={10,20,30,40,50} which is A

(A’)’ = A

It is clear from the definition of the complement set that for any subset of the universal set U, we have

(A’)’ = A

Set Operations

- Difference of Sets

Consider, A {1, 2, 3, 4, 5, 6) and B = {2, 3, 5, 7, 9}

These sets are represented by a Venn diagram.

What does the shaded portion indicate?

The shaded portion represents a new set of elements {1,4,6}

This set is formed by the elements of set A, excluding those elements which are present, in set B.

This new set is the difference of set A and set B. i.e. A – B

Set Operations – Difference

The set A – B denotes a set which is formed by taking elements of A which do not belong to B.

Therefore A – B = {1, 4, 6}

Similarly, We can find the difference of sets B and A i.e. B – A

The set B – A is formed by taking elements of B which do not belong to A.

Therefore B – A = {7,9}

Complement and Difference - Exercises

- Given U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 2, 3, 4} find the complement of A.
- If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} find the complement of the following sets:
- A = {2, 4, 6, 8} 2. B = {1, 3, 5, 7, 9}

3. C = {2, 3, 5, 7} 4. D = 5. U

- Find P’, if U = {all natural numbers less than 20} and P = {all prime numbers less than 10}
- Find A’ if U = {Number of students in your class} and A = {Number of boys in your class}.

Complement and Difference - Exercises

- Let U = {a, b, e, d, e, f, g, h, i, j, k} A= {a, b, c, d} and B = {a, d, g, j} find

1. (A B)’ 2. (A B)’ 3. A’ B’ 4. A’ B’

- By taking a suitable example, show that
- ‘= U
- U’ =
- Let H = {1, 4, 9, 16, 25} and J = {I, 2, 4, 6, 8, 16, 32} find i) H – J ii) J – H
- If A= {3, 4, 5}, B={4, 5, 6} and C = {5, 6, 7, 8} find

i) A-B ii) B-C iii) C-A iv) B-A v) C-B

Complement and Difference - Exercises

- Show that A’ – A =
- Let U= {a, b, c, d, e, f, g}A = { a, b, c, d, e}B= {a, c, e, g} and C= {b, e, f, g} then findi) (C – B)’ ii) (A – C)’ iii) (A – B)’iv) (B – C)’ v) (C – A)’

Unit 5 – Matrices

- Objectives
- Recall the definition of matrix
- Classify the matrices based on their elements and order

Recalling Matrix

- A matrix is a rectangular arrangement of numbers in rows and columns, enclosed within brackets
- Example:
- P =
- The order of this matrix is 2 x 3
- 1, 2, 4, 5, 3 and 9 are its elements.

Matrices - Types of Matrices

- We can classify the matrices into different types based on the elements they contain and their order.
- First let us study the classification of the matrices based on their order:

You know that in a matrix if there are ‘m’ rows and ‘n’ columns then

its order is m x n.m x n is read as m cross n.

Matrices

- What is the relationship between the number of rows and columns in each matrix?

A B C D

- Among the first two matrices, the number of rows is equal to the number of columns. But it is not so in the case of other two matrices.
- In the above examples, matrices A and B are square matrices and matrices C and Dare rectangular matrices.

Matrices – Examples

- In the square matrix A, observe the elements shown by dotted lines. They are principaldiagonal elements.
- Therefore In a square matrix, elements from left top comer to the right bottom comer are called the principal diagonal elements.
- In this square matrix, the principal diagonal elements are 1, 8 and 6.

Types of Square Matrices

- Diagonal Matrix

Consider the following matrices A & B.

Observe their elements. Do you find any specialty?

In these square matrices, excluding the principal diagonal elements, all the other elements are equal to zero, such a matrix is called a diagonal matrix.

A square matrix in which all the elements are zero, except the principal diagonal elements, is called a diagonal matrix.

Types of Square Matrices

- Scalar Matrix

Study the matrices given below. Identify the principal diagonal elements and compare them. What do you find?

In these diagonal matrices, the principal diagonal elements are equal to one another. Such a diagonal matrix is called a scalar matrix.

A diagonal matrix is a scalar matrixin which the principaldiagonal elements are equal.

Types of Square Matrices

- Unit Matrix or identity matrix

Study the matrices given below. Observe the principal diagonal.

In these matrices the principal diagonal elements are equal to 1. Such a matrix is known as Unit matrix orIdentity matrix.

It is denoted by . Unit Matrix is also a Scalar Matrix.

A diagonal matrix in which each diagonal element is equal tounity is called a unit matrix or identity matrix.

Types of Square Matrices

- Symmetric Matrix

Consider the square matrices M and N. Draw a line passing through the diagonal elements. Imagine that you are going to fold the matrix along the diagonal. Observe the overlapping elements. What do you find?

We observe that, the overlapping elements are equal to each other. Such matrices are called symmetric matrices.

A square matrix is called a symmetric matrix if the elements which are symmetric with respect to the principal diagonal are equal.

Types of Square Matrices

- Skew Symmetric Matrix

Consider the square matrices C and D. Observe their principal diagonal elements and the signs of the symmetric elements. What do you find?

We find that. The principal diagonal elements are equal to zero. The elements which are symmetric with respect to the principal diagonal are equal but opposite in sign. Such square matrices are called skew symmetric matrices.

A square matrix is called a skew symmetric matrix, if the elements which are symmetric with respect to the principal diagonal are equal but opposite in sign, while the Principal diagonal elements are zero

Types of Rectangular Matrices

- Row Matrix

Observe the following rectangular matrices.

Find out the order of each matrix.

What is the common feature among them?

A=[1 4 6] B=[4 19]

C=[-1 0 8 9] D=[a b c d e]

The above rectangular matrices have only one row irrespective of the number of columns. Such matrices are called row matrices.

A matrix which has only one row is called a row matrix.

Its order is 1 x n

Types of Rectangular Matrices

- Column Matrix

Observe the following rectangular matrices.

Find out the order of each matrix. What is the common feature among them?

All the above matrices have only one column irrespective of the number of rows. Such matrices are known as column matrices.

A matrix which has only one column is called a column matrix.

Its order is m x 1

Types of Rectangular Matrices

- Zero Matrix

Now we shall study a special kind of matrix, which may be either a square matrix or a rectangular matrix. But we define it on the basis of its elements only. Study these matrices.

S is a square matrix and T is a rectangular matrix. Though different types of matrices, they have a special property in common. We find all the elements are equal to zero. Such a matrix having all its elements equal to zero is called zero matrix or null matrix.

Zero matrix is a matrix in which every element is equal to zero.

Matrix Properties

- Equality

Observe the following matrices. Compare their orders and also the corresponding elements.

We find that their orders are equal and the corresponding elements are also equal. Such matrices are said to be equal to each others. A = B

Two matrices are said to be equal if and only if they are of the same order and each element in the first matrix is equal to the corresponding element in the second matrix.

Matrix Properties

- Equality

If and only if p = 1, q = 2, r = 3 and s = 4

Equality of two matrices can be used to find any unknown element in one of them.

Equality of Matrices - Examples

- Find x if

This is a matrix equation. Since the matrices are equal, the corresponding elements are equal.

x = 6

- Solve x if A = B and

Since A = B, x + 6 = 15, (as corresponding elements are equal

x = 15 – 6 = 9 Therefore x = 9

Matrix Properties

- Transpose of a Matrix

Consider

Let us rewrite rows of A as corresponding columns. We get a new matrix.

A’ is called the transpose of A.

Here the order of matrix A is 2 x 3, and that of A’ is 3 x 2.

Transpose of a matrix is the matrix obtained on interchanging its rows and columns.

Transpose of Matrices - Examples

- Let us study some more examples

Thus, if the given matrix is of the order m x n then it’s transpose is of order in n x m.

Transpose of Matrices - Examples

Thus for any matrix A : (A ‘) ‘ = A

Try!Interchange the rows and columns of a symmetric matrix. What do you get?

Matrices - Exercises

- Identify the type of the following matrices:

Matrices - Exercises

- Write the principal diagonal elements of these matrices.

Matrices - Exercises

- Give One example for each of the following:
- Square matrix of order 2 x 2
- Diagonal matrix of order 3 x 3
- Rectangular matrix of order 4 x 2
- Row matrix of order 1 x 5
- Scalar matrix of order 4 x 4
- Identity matrix of order 5 x 5
- Skew symmetric matrix of order 3 x 3
- Symmetric matrix of order 4 x 4
- Column matrix of order 6 x 1
- Null matrix of order 6 x 6

Matrices - Exercises

- Solve the matrix equations:

Matrices - Exercises

- Find the Transposeof the following matrices:

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