Subject: Optional Mathematics
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x,y) is in the relation. A technical definition of a function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. This means that if the object x is in the set of inputs (called the domain) then a function fh
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.
Laxman is younger brother of Ram; Arjun is younger brother of Bhimsen; Dhirendra is younger brother of Pravin. Draw a mapping diagram to show the relationship from younger to elder.
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.
If A= {4,2,36,24} and B= {1,2,18,12}, draw a mapping diagram to show the relationship 'double of from A to B.
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.
Ranju and Anju are tall. Ranju and Bharat are black. Bharat and Radha are short. Draw a mapping diagram to show the above characteristics of Ranju, Anju, Bharat and Radha. Answer the following questions on the basis of the mapping diagram:
(a) Who are tall and black?
(b) Who are short and Black?
(c) Write down the characteristics of Anju and Radha?
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.
By studying the diagram (b) of question 9, answer the following questions.
(a) What is the common factor of both 2 and 3?
(b) What is the factor of 1?
(c) What are the factors of 2?
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.
If A= {3,4,5,6,7} and B= {1,2,3,4} draw the mapping diagram to show the relationship 'less than 2' from A and B.
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
If A={Shanti,Geeta,Sita} and B={Radio,T.V} then find B×A.
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.
Express the relationship between group P and Q by a description from the mapping diagram given below.
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.
Find A×B from the mapping diagram below and show on a graph.
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:
Draw arrow to show the relationship 'is the capital of' from group A to group B.
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.
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