Write the smallest reflexive relation on set {1, 2, 3, 4}.
Last updated date: 16th Mar 2023
•
Total views: 306.3k
•
Views today: 5.85k
Answer
306.3k+ views
Hint: The smallest reflexive relation on set (a, b, c, d) is { (a, a), (b, b), (c, c), (d, d) }. To find the reflexive relation on a given set, take a = 1, b = 2, c = 3 and d = 4 in this question.
We are given a set {1, 2, 3, 4} and we have to find the smallest reflexive relation the given set.
Before proceeding with this question, we will see what a “set” is.
A set in mathematics is a collection of well- defined and distinct objects, considered as an object in its own right. The most basic property is that a set “has” elements.
Like, we are given a set {1, 2, 3, 4} whose elements are 1, 2, 3 and 4.
Now the relations on sets are nothing but the properties of sets. We have basically three types of relations on set - that are,
1) Reflexive relation
2) Transitive relation
3) Symmetric relation
Now, we know that reflexive relation is the relation in which each element of a set is related to itself.
That means if we have set {a, b, c, d} then the smallest reflexive relation on this set is { (a, a), (b, b), (c, c), (d, d) }.
This is the smallest relation because this relation does not contain any other relation on set apart from reflexive relation.
Therefore for set {1, 2, 3, 4},
We have,
a = 1
b = 2
c = 3
d = 4
Therefore, the smallest reflexive relation the given set is { (1,1), (2, 2), (3, 3), (4, 4) }.
Note: Students should keep in mind that a solution on set A is reflexive only when the relation contains each and every element of A related to itself. The relation must not leave any element of set A.
Here we have some examples of reflexive relation that are as follows:-
- “is equal to” (equality)
- “is a subset of” (set inclusion)
- “divides” (divisibility)
- “is greater than or equal to”
- “is less than or equal to”
Now we also have some examples of relations that are irreflexive that are as follows:
- “is not equal to”
- “is coprime to” (for integers > 1, since 1 is coprime to itself)
- “is a proper subset of”
- “is greater than”
- “is less than”
We are given a set {1, 2, 3, 4} and we have to find the smallest reflexive relation the given set.
Before proceeding with this question, we will see what a “set” is.
A set in mathematics is a collection of well- defined and distinct objects, considered as an object in its own right. The most basic property is that a set “has” elements.
Like, we are given a set {1, 2, 3, 4} whose elements are 1, 2, 3 and 4.
Now the relations on sets are nothing but the properties of sets. We have basically three types of relations on set - that are,
1) Reflexive relation
2) Transitive relation
3) Symmetric relation
Now, we know that reflexive relation is the relation in which each element of a set is related to itself.
That means if we have set {a, b, c, d} then the smallest reflexive relation on this set is { (a, a), (b, b), (c, c), (d, d) }.
This is the smallest relation because this relation does not contain any other relation on set apart from reflexive relation.
Therefore for set {1, 2, 3, 4},
We have,
a = 1
b = 2
c = 3
d = 4
Therefore, the smallest reflexive relation the given set is { (1,1), (2, 2), (3, 3), (4, 4) }.
Note: Students should keep in mind that a solution on set A is reflexive only when the relation contains each and every element of A related to itself. The relation must not leave any element of set A.
Here we have some examples of reflexive relation that are as follows:-
- “is equal to” (equality)
- “is a subset of” (set inclusion)
- “divides” (divisibility)
- “is greater than or equal to”
- “is less than or equal to”
Now we also have some examples of relations that are irreflexive that are as follows:
- “is not equal to”
- “is coprime to” (for integers > 1, since 1 is coprime to itself)
- “is a proper subset of”
- “is greater than”
- “is less than”
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
