Let A and B be two sets. Then, A×B is called the Cartesian product of A and B. Any subset of A×B is called a relation from A to B. A relation from A to B is denoted by R:A→B or simply by R. For example,
Let A = {1,2} and B = {3,4}
Then, AB = {(1,3),(1,4),(2,3),(2,4)}
Let, R = {(1,3),(2,3)}
Here, R is a subset of A×B. So, R is a relation in A.
A relation from set A to set B is defined by a subset of Cartesian product A× B under the certain condition. The relation is generally denoted by R: A → B.
The first set A is called Domain of relation.
Second set B is called range of the relation.
Relation may be expressed in various ways. Some ways are:
a. Listing or ordered pair form
R= {(Hari,Sujit)(Uma,Sharmila)(Sagarmatha,Dhaulagiri)}
R = {(1,1),(2,2),(3,3)}
b. Table form
Let, A = {1,2,3} and B = {1,4,3}.
Define R = {(x,y):y=x^{2}}
Then, the relation R can be represented as:
x | 1 | 2 | 3 |
y | 1 | 4 | 9 |
c. Set builder form
Let A = {1,2,3} and B = {4,5,6} , Define R = {(x,y):x=y} i.e. R= {(1,1),(2,2),(3,3)}
d. Graphical form
Let A = {1,2,3} and B = {1,2,3}, R={(x,y):y=x^{2}}
e. Arrow diagram form
Let A = {a, b} and B = {1, 2}
If R = {(a,b): a∈A and b∈B} is a relation, then there exists another relation {(b, a) b∈B and a∈A} which is known as the inverse relation of R.
it is denoted by R^{-1}.
Mathematically, R^{-1}= [(b,a) : b∈ B and A ∈ a}
For example, If R = {( 1,a), (2,b), (3,c)}, then
R^{-1}= {(a,1), (b,2), (c,3)}
The first set A is called Domain of relation. Second set B is called range of the relation.
soln:
here set of younger brother ={Laxman,Arjun,Dhirendra}
Set of elder brother= {Ram,Bhimsen,Pravin}
The mapping diagram to show the relationship from younger to elder brother is as shown below:
∴ Relation of younger and Elder brother = {(Laxman,Ram), (Arjun,Bhimsen), (Dhirendra, Pravin)} Ans.
Soln:
Here given sets are:
A= {4,2,36,24}, B={1,2,18,12}
The mapping diagram from A to B show the relationship "Double of" is as shown below:
∴ "Double of " relation= {(4,2),(2,1)(36,18),(24,12)} Ans.
Soln:
here given,
Set of persons = { Ranju,Anju,Bharat,Radha}
Set of characteristics = {tall,black,short}
The mapping diagram from set of persons to set of characteristics is as shown below:
From the above mapping diagram:
(a) Ranju is tall and black
(b) Bharat is short and black
(c) Anju is tall and Radha is short. Ans.
Soln:
From figure in Q.9 (b)
(a) Since 2 and 3 both are related to 1, so 2 and 3 have a common factor 1. Ans.
(b)Since 1 is related to 1 only. So, common factor of 1 is 1. Ans.
(c) Since 2 is related to 1 and 2. So the common factors of 2 are 2 and 1. Ans.
soln:
here given,
A={3,4,5,6,7} and B={1,2,3,4}
The mapping diagram from A to B to show the relationship 'less than 2' is as shown below:
Here each number of B is less than 2 then the number from A.
∴ Relation= {(3,1),(4,2),(5,3),(6,4). Ans
Soln:
here given,
A= {Shanti,Geeta,Sita}, B={Radio,T.V}
Now,
B×A= {Radio,T.V}×{Shanti,Geeta,Sita}
={(Radio,Shanti),(Radio,Geeta),(Radio,Sita),(T.V,Shanti),(T.V, Shanti),(T.V,Geeta),(T.V,Sita)} Ans.
Soln:
(a) Here given arrow diagram:
Relation between P and Q={(1,1),(2,4)(3,9)} which is the square relation.Ans
(b) Here given arrow diagram:
Relation between P and Q= {(1,1),(1,2),(1,3),(2,2),(3,3)} which is the less or equal to relation. Ans.
Soln:
∴ A×B= {(3,4),(3,5),(3,6),(3,7),(4,4),(4,5),(4,6),(4,7),(5,4),(5,5),(5,6),(5,7),(6,4),(6,5),(6,6),(6,7)}. Ans.
The mapping diagram of A×B is given below:
∴ A×B={(3,4),(3,5),(3,6),(3,7),(4,4),(4,5),(4,6),(4,7),(5,4),(5,5),(5,6),(5,7),(6,4),(6,5),(6,6),(6,7)}. Ans.
The graph of A×B is given below:
Soln:
The arrow diagram From A to B to show relationship "is the capital of " is as follow:
Here, first set A is the capital and second set B is the country. So first is the capital second.
∴ A×B={(Thimpu,Bhutan),(Tokyo,Japan),(Beijing,China),(Dhaka,Bangaladesh)}. Ans.
For what value of x and y, (x+5,y+2) and (7,5) are equal to each other?
2,3
2,4
2,5
1,2
For what value of x and y, (x+y,1) and (2,2x-y) are equal to each other?
1,2
1,3
2,2
1,1
For what values of x andy,(x-2,y-3)and (2x-6,3)are equal to each other ?
4,6
4,4
3,7
2,6
If A xB ={(1,2),(2,3),(1,3),(2,2),(3,2),(3,3)}, Find A and B.
A={1,5,3},B ={2,3}
A={1,2,1},B ={2,3}
A={1,2,3},B ={2,3}
A={1,2,3},B ={1,3}
Find the values of x and y.
(a) (3x,2y + 2)=(12,3y-1)
Find the values of x and y.
(b) (2x-1,y+2)=(-1,2)
Find the values of x and y.
(c) (2x+4,y+5)=(3x+3,6)
(6,1)
(1,4)
(1,1)
(2,2)
Find the values of x and y.
(d) (3x-6,y+2)=(x-2,4)
(2,3)
(1,1)
(2,1)
(2,2)
Find the values of x and y.
(x+y, x-y)=(8,0)
(5,4)
(4,0)
(4,4)
(4,2)
If R = {(x,y):y=2x} and x ∈{2,4,6},find the range of R.
{4,6,12}
{1,2,12}
{4,8,12}
{2,8,12}
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