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Just like in life, figuring out relationships in Mathematics makes the path clearer. The NCERT solutions for Exercise 1.1 in Class 12 Maths Chapter 1, Relations and Functions, help us untangle that web. A relation is like a phone contact list where each person can have one or more numbers connected to them, but Functions are that special kind of contact list where each person has only one number to contact, nothing else. In Exercise 1.1 of Class 12 Maths Chapter 1 of the NCERT, we encounter several problems related to various kinds of relations, including reflexive, symmetric, transitive, and equivalence relations.
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Experienced Careers360 subject matter experts have made these NCERT solutions by explaining every step and providing the necessary concepts. Before jumping into the functions exercise, it is necessary to master these concepts about relations.
Question1(i) . Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation
Answer:
Since
Since
Since
Hence,
Question 1(ii) . Determine whether each of the following relations are reflexive, symmetric and
transitive:
(ii) Relation R in the set N of natural numbers defined as
Answer:
Since,
so
Since,
so
Since there is no pair in
Hence,
nor transitive.
Question1(iii) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iii) Relation R in the set
Answer:
Any number is divisible by itself and
Hence, it is transitive.
Hence, it is reflexive and transitive but not symmetric.
Question.1(iv) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iv). Relation R in the set Z of all integers defined as
Answer:
For
So,it is reflexive.
For
So, it is symmetric.
For
Now,
So,
Hence, it is reflexive, symmetric and transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(a)
Answer:
Hence, it is reflexive, symmetric and transitive.
Question 1 (v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(b)
Answer:
It is same as
So,it is symmetric.
It implies that
Hence, it is reflexive, symmetric and transitive.
Question 1 (v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(c)
Answer:
Hence, it is not reflexive, symmetric, or transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v). Relation R in the set A of human beings in a town at a particular time given by
(d)
Answer:
So, it is not reflexive.
So, it is not symmetric.
Let,
This case is not possible, so it is not transitive.
Hence, it is not reflexive, symmetric or transitive.
Question 1 (v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(e)
Answer:
If
But a person cannot be their own father, so
If
This does not imply that
Suppose
Then
This means
So,
So, it is not transitive.
Hence, it is neither reflexive nor symmetric nor transitive.
Answer:
Taking
and
So, R is not reflexive.
Now,
But,
So,
Hence, it is not symmetric.
Since
Hence, it is not transitive.
Thus, we can conclude that it is neither reflexive, nor symmetric, nor transitive.
Question:3 Check whether the relation R defined in the set
Answer:
Since
So, it is not transitive.
Hence, it is neither reflexive, nor symmetric, nor transitive
Question:4 Show that the relation R in R defined as
Answer:
As
Now we take an example
But
So,it is not symmetric.
Now if we take,
Than,
So, it is transitive.
Hence, we can say that it is reflexive and transitive but not symmetric.
Question:5 Check whether the relation R in R defined by
symmetric or transitive.
Answer:
So, it is not symmetric
Now,
but
It is not symmetric
But,
So it is not transitive
Thus, it is neither reflexive, nor symmetric, nor transitive.
Question:6 Show that the relation R in the set
symmetric but neither reflexive nor transitive.
Answer:
Let A=
We can see
As
But
Hence, R is symmetric but neither reflexive nor transitive.
Answer:
A = all the books in a library of a college
Let
Since y and x have the same number of pages, so
Hence, it is symmetric.
Let
and
This states,x and z also have the same number of pages i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive, i.e. it is an equivalence
relation.
Answer:
Let there be
Let
Hence, it is symmetric
Now, let
then,
So,
Thus, it is reflexive, symmetric and transitive, i.e. it is an equivalence relation.
The elements of
The elements of
The element of
Question:9(i) Show that each of the relation R in the set
(i)
Answer:
For
Henec, it is reflexive.
Let,
then
Hence, it is symmetric.
Let,
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is
Question:9(ii) Show that each of the relation R in the set
(ii)
Answer:
For
Henec, it is reflexive.
Let,
Hence, it is symmetric.
Let,
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is {1}
Question:10(i) Give an example of a relation.
(i) Which is Symmetric but neither reflexive nor transitive.
Answer:
Let
Hence, symmetric but neither reflexive nor transitive.
Question:10(ii) Give an example of a relation.
(ii) Which is transitive but neither reflexive nor symmetric.
Answer:
Let
Now for
Let
Then
Let
we can write this as
Hence,
Hence, it is transitive but neither reflexive nor symmetric.
Question:10(iii) Give an example of a relation.
(iii) Which is Reflexive and symmetric but not transitive.
Answer:
Let
Define a relation R on A as
If
If
But
Hence, it is Reflexive and symmetric but not transitive.
Question 10 (iv) Give an example of a relation.
(iv) Which is Reflexive and transitive but not symmetric.
Answer:
Let there be a relation R in R
Let
But
So it is not symmetric.
Let
This can be written as
Hence, it is transitive.
Thus, it is Reflexive and transitive but not symmetric.
Question 10 (v) Give an example of a relation.
(v) Which is Symmetric and transitive but not reflexive.
Answer:
Let there be a relation A in R
We can see
So it is symmetric.
Let
Also
Hence, it is transitive.
Thus, it Symmetric and transitive but not reflexive.
Answer:
The distance of point P from the origin is always the same as the distance of same point P from origin i.e.
Let
this is the same as distance of the point Q from the origin is the same as the distance of the point P from the origin i.e.
Let
i.e. the distance of point P from the origin is the same as the distance of point Q from the origin, and also the distance of point Q from the origin is the same as the distance of the point S from the origin.
We can say that the distance of point P, Q, S from the origin is the same. Means distance of point P from the origin is the same as the distance of point S from origin i.e.
Hence, R is an equivalence relation.
The set of all points related to a point
In other words, we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from the origin.
Hence, these sets of points form a circle with the centre as the origin and this circle passes through the point.
Answer:
All triangles are similar to itself, so it is reflexive.
Let,
T 1 is similar to T2 is the same asT2 is similar to T 1 i.e.
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
Now, we see the ratio of sides of triangle T 1 andT 3 are as shown
i.e. ratios of sides of T 1 and T 3 are equal.Hence, T 1 and T 3 are related.
Answer:
The same polygon has the same number of sides with itself,i.e.
Let,
P 1 have the same number of sides as P 2 is the same as P 2 have same number of sides as P 1 i.e.
Hence,it is symmetric.
Let,
Hence, it is transitive,
Thus,
The elements in A related to the right angle triangle T with sides 3, 4 and 5 are those polygons which have 3 sides.
Hence, the set of all elements in A related to the right angle triangle T is set of all triangles.
Answer:
All lines are parallel to itself, so it is reflexive.
Let,
L1 is parallel to L 2 is same as L 2 is parallel to L 1 i.e.
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
The set of all lines related to the line
Here, Slope = m = 2 and constant = c = 4
It is known that the slope of parallel lines are equal.
Lines parallel to this (
Hence, set of all parallel lines to
Question:15 Let R be the relation in the set A= {1,2,3,4}
given by
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.
Answer:
A = {1,2,3,4}
For every
Given,
For
Hence, R is reflexive and transitive but not symmetric.
The correct answer is option B.
Question 16 Let R be the relation in the set N given by
(A)
(B)
(C)
(D)
Answer:
(A) Since,
(B) Since,
(C) Since,
(d) Since,
The correct answer is option C.
Also Read,
Types of Relation | Definition | Example |
Empty Relation | No element in the set is related to any other element. | Set A = {1, 2}, R = ∅ (no pairs like (1,1) or (2,2)) |
Universal Relation | Every element is related to every element in the set. | A = {a, b}, R = {(a,a), (a,b), (b,a), (b,b)} |
Identity Relation | Each element is related only to itself. | A = {1, 2, 3}, R = {(1,1), (2,2), (3,3)} |
Inverse Relation | Flips the order of elements in a relation. | R = {(1,2), (3,4)} → R⁻¹ = {(2,1), (4,3)} |
Reflexive Relation | Every element is related to itself. | A = {x, y}, R = {(x,x), (y,y)} |
Symmetric Relation | If (a, b) ∈ R, then (b, a) must also be in R. | R = {(1, 2), (2, 1)} |
Transitive Relation | If (a, b) and (b, c) are in R, then (a, c) must also be in R. | R = {(1, 2), (2, 3), (1, 3)} |
Also, read,
These are links to other subjects' NCERT textbook solutions. Students can check and analyse these well-structured solutions for a deeper understanding.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
Students can check these NCERT exemplar links for further practice purposes.
Concepts related to symmetric, reflexive and transitive, equivalence relations etc, are discussed in the Exercise 1.1 Class 12 Maths
Topics like
Two chapters 'relation and function' and 'inverse trigonometry' combined has 10 % weightage in the CBSE final board exam.
From the analysis of previous year questions of Board exams, it is clear that direct questions are asked from the NCERT questions. Also Some of the questions are repeated year after year. Hence it is said that NCERT solutions are low hanging fruits. Every serious student must practice NCERT questions to score well in the exam.
In maths, relation defines the relationship between sets of values of ordered pairs
symmetric, reflexive and transitive, equivalence relations etc, are discussed in the Exercise 1.1 Class 12 Maths
There are 15 questions in Exercise 1.1 Class 12 Maths
In NCERT class 12 maths chapter 1 relations and function, there are a total of 5 exercises which includes a miscellaneous exercise also.
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Hello there! Thanks for reaching out to us at Careers360.
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If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
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