Subject: Optional Maths

As we know in mathematics order plays an important role .In some case we may feel not so important for example,{Shiva,Parvati} or {Parvati, Shiva} as both are sets of two people but here if we start thinking of a pair of gloves then right side of gloves should be usedin right and left side of glove should be used in left . If ordered of glove change than it is not comfortable at all.Hence ordered plays a vital role.

**Order**

An order is an arrangement of members or elements in a meaningful way is called order . Some example of order are;

(1)5,6,7,8,9

It is an order of numbers whose each next number is 1.

(2)a,b,c,d,e

This is a first five English alphabets an order.

(3)2,4,6,8,10

first five multiples of 2 in ascending order.

**Pair**

A pair is a combination of two objects or number . for example,a pair of gloves ,a pair of shoes ,Shiva and Parvati,Nepal and Kathmandu etc

**Ordered pair**

The combination of numbers or elements in a fixed order is called ordered pair.Here the first elements represent the x-component (x) and the second elements represent the y-components (y).An ordered pair are always separate by comma and closed by small brackets. For example (a, b), (1, 2), (Ram, Sita) etc.

Here in order pair the first x and the second y component should always follow the order if not then it can give different meaning .

**For example ;**

The ordered pair bellow is a pair which represents the nation and its capital city

(Nepal, Kathmandu)

(India, New Delhi)

But if we write (Kathmandu,Nepal), Kathmandu becomes the nation and Nepal become the capital city which is completely wrong . So we need to follow in a meaningful order here in case of set {Nepal, Kathmandu } or {Kathmandu, Nepal} both are same. but here its plays an important role.

The x and y component is known as antecedent and consequence respectively .We can also use a diagram to represent the ordered pairs.

**Equality of ordered pair **

The ordered pairs (e, f)and (g, h) are said to be equal if their corresponding elements are equal . i.e, the first component **e**should be equal to the first components **g**and the second component should be equal to the second component**.**

**Mathematically,**

If ( e, f ) = ( g, h ) then

e = g and f = h

For example; Is the ordered pair equal ?

(5, 6) = (5, 6)

Hence, the ordered pair x=5 and y =6 in both.

**Cartesian product**

Let M and N be two non-empty sets . Then the products M * N read as M cross N is defined as the set of all ordered pairs in such a way that the first elements is taken from set M and the second element is taken from set B.

**Mathematically**

M ×N = {(x, y) :x∈**M and **y∈**N**}

Now , M = {1, 2} N = {a, b }

Then, M × N can be calculated as in the following table.

Now, M ×N in the set of all above ordered pairs and we write it as ,

M ×N = { ( 1. a), ( 1, b), ( 2,a ), (2, b) }

Again, N × M can be written as,

B

A

a | b | |

1 | (1, a) | (1, b) |

2 | (2, a) | (2, b) |

So , N × M = { ( a, 1), ( a, 2 ), (b, 1), (b, 2) }

from example also it is observed that

since, (1, a ) ≠ ( a, 1 )

M * N ≠ N × M

The cartesian product can be also obtained by using an arrow diagram .

from this arrow diagram also we can see that

M × N = { (1, a ), (1, b ), ( 2, a ), ( 2, b ) }

Again, for N × M , we place N first and M second and draw arrows to represent it.

So, N×M = {(a, 1 ), ( a, 2 ), (b, 1 ), ( b, 2)}

figure ;

- Order is the arrangement of member or elements in a sequential manner or a meaningful way .
- Pair is the combination of two objects or numbers.
- For eg ; ( p , q ) and ( q , p ) are not equal ordered pairs as in ( p, q ) the first component is p and the second component is q and vice versa in ( q , p ).

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

What do you mean by order ?

An order is an arrangement of members or elements in a meaningful way is called order .

What is an ordered pair ?

The combination of numbers or elements in a fixed order in such a way that the first elements always represent the x-component (x) and the second elements always represent y-components (y) is known as an ordered pair.

What do you mean by the pair ?

The combination of two objects or numbers is known as a pair .

What is an ordered pair of ( 1, 2 ) ?

In an ordered pair ( 1, 2 ) , 1 is the first component and 2 is the second component .

Is the ordered pair equal ?

( 3, 2 ) = ( 1, 2 )

No, the ordered pair is not equal .

Find the values of x and y.

(x, y) = (2, 5)

Solution

(x, y) = (2, 5)

Since the ordered pairs are equal, the corresponding elements are also equal.

So, x = 2 and y = 5

Find the value of x and y.

(x - 2, 2) = (2, y +1)

Solution

Since the ordered pairs are equal the corresponding elements are also equal.

So, x - 2 = 2

x = 2 + 2

x = 4

Similarly, 2 = y + 1

y + 1 = 2

y = 2 - 1

y = 1

Hence, x = 4 and y = 1

Find the value of x and y.

(x - 7, 5) = (5, y +1)

Solution

Since the ordered pairs are equal the corresponding elements are also equal.

So, x - 7 = 5

x = 5 + 7

x = 12

Similarly, 5 = y + 1

y + 1 = 5

y = 5 - 1

y = 4

Hence, x = 12 and y = 4

Find the value of x and y.

(x - 2, 10) = (4, y +5)

Solution

Since the ordered pairs are equal the corresponding elements are also equal.

So, x - 2 = 10

x = 10 + 2

x = 12

Similarly, 4 = y + 5

y + 5 = 4

y = 4 - 5

y = -1

Hence, x = 12 and y = - 1

If A =(3, 4) and B =(5, 6)

Find A × B and B × A

Solution

Here, A= (3, 4) and B(5, 6)

Now, A × B is given by,

So, A × B = {(3, 4) (3, 6) (4, 5) (4, 6)}

Again, B × A = { ( 5, 3) (5, 4) (6, 3) (6, 4) }

If P×Q = { (3, 4) (5, 5) (5, 6) (3, 5) }. Find set P and Q.

Solution

Here, P × Q = { (3, 4) (5, 5) (5, 6) (3, 5) }

According to the definition of cartesian product P × Q, the first element should be from set P and the second element should be from set Q.

So, Set P contain the first elements that is 3, 5, 5, 3. But, in the set the elements should not be repeated.

Hence, P = { 3, 4}

Similarly, Set Q should contain all the y components that is. { 4, 5, 6, 5}

Again, Q = {3, 5}

P = {3, 4}

Q = {3, 5}

If A = { 6, 8 } B = {6, 7} C = {8, 0}. Find A × B and A × C.

Solution

Here, A = { 6, 8 } B = {6, 7} C = {8, 0}

Now, A × B in the set of all above ordered pairs and we write it as,

A × B = { ( 6, 6) (6, 7) (8, 6) (8, 7) }

Similarly,

A × C = { {6, 8) (6, 0) (8, 8) (8, 0) }

If P = { 5, 6 } Q = {4, 7} R = {6, 2}. Find P × Q and P × R.

Solution

Here, P = { 5, 6 } Q= {4, 7} R = {6, 2}

Now, P × Q in the set of all above ordered pairs and we write it as,

P × Q = { ( 5, 4) (5, 7) (6, 4) (6, 7) }

Similarly,

P × R = { {5, 6) (5, 2) (6, 6) (6, 2) }

Find the value of x and y.

( 3x - 5, 0) = (x -2, y +3 )

Solution

Given information,

( 3x - 5, 0) = (x -2, y +3 )

Since the ordered pairs are equal, the corresponding elements are also equal.

So, 3x - 5 = x - 2

or, 3x - x = -2 + 5

or, 2x = 3

or, x = \(\frac{3}{2}\)

Again, 0 = y + 3

or, y + 3 = 0

or, y = -3

Hence, x = \(\frac{3}{2}\) and y = -3

If A = {3, 4} and B = { 1, 2}. Find A × B.

Solution

Here, A ={3, 4} and B = {1, 2}

Now, A × B = {(3, 1) (3, 2) (4, 1) (4, 2) }.

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