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Understanding relations and functions in mathematics is like learning to navigate a map – it helps us move from one point to another with purpose. During the NCERT Solutions of relations and functions class 12, the first question that comes to mind is, what exactly are relations and functions? A relation is like a connection between a student and all the subjects in the syllabus. On the other hand, a function is like a special subject, which is the student's only favourite subject. The main purpose of these relations and functions class 12 NCERT solutions is to make learning easier for students and to explain this topic more easily.
Few months back, the Central Board of Secondary Education (CBSE) has made some changes in the curriculum and textbook for applied mathematics and has also plans of allowing the use of calculator in the CBSE Class 12 accounts subject.
People use relations and functions in real-life applications, such as mapping, programming, and data handling. These solutions by the NCERT are essential for students because they comprise quality practice questions. After checking the Class 12 Maths NCERT solutions from textbooks, students can also check the NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions to understand this chapter better.
Students who wish to access the Class 12 Maths Chapter 1 NCERT Solutions can click on the link below to download the complete solution in PDF.
>> Relations:
A relation
- Domain of
- Range of
>> Functions:
A relation
If
A function
Algebra of functions:
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NCERT Relations and Functions Class 11 Solutions: Exercise: 1.1 Page Number: 5-7 Total Questions: 16 |
Question 1(i): Determine whether each of the following relations are reflexive, symmetric, and transitive:
(i) Relation
Answer:
A={1,2,3...,13,14}
R={(x,y):3x−y=0} ={(1,3),(2,6),(3,9),(4,12)}
Since, (1,1),(2,2),(3,3),(4,4),(5,5)⋅⋅⋅⋅⋅⋅(14,14)∉R so R is not reflexive.
Since (1,3)∈R but (3,1)∉R, R is not symmetric.
Since, (1,3),(3,9)∈R but (1,9)∉R, so R is not transitive.
Hence, R is neither reflexive nor symmetric nor transitive.
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:
R={(x,y):y=x+5andx<4} ={(1,6),(2,7),(3,8)}
Since (1,1)∉R
So R is not reflexive.
Since, (1,6)∈R but (6,1)∉R
So R is not symmetric.
Since there is no pair in R such that (x,y),(y,x)∈R so this is not transitive.
Hence, R is neither reflexive nor symmetric nor transitive.
Question 1 (iii): Determine whether each of the following relations is reflexive, symmetric, and Transitive:
(iii) Relation R in the set A={1,2,3,4,5,6} as R={(x,y):y is divisible by x}
Answer:
A={1,2,3,4,5,6}
R={(2,4),(3,6),(2,6),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
Any number is divisible by itself, and (x,x)∈R. So it is reflexive.
(2,4)∈R but (4,2)∉R .Hence,it is not symmetric.
(2,4),(4,4)∈R and 4 is divisible by 2 and 4 is divisible by 4.
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 R={(x,y):x−y is an integer}
Answer:
R={(x,y):x−y is an integer}
For x∈Z, (x,x)∈R as x−x=0, which is an integer.
So, it is reflexive.
For x,y∈Z, (x,y)∈R and (y,x)∈R because x−y and y−x are both integers.
So, it is symmetric.
For x,y,z∈Z, (x,y),(y,z)∈R as x−y and y−z are both integers.
Now, x−z=(x−y)+(y−z) is also an integer.
So, (x,z)∈R and hence it is transitive.
Hence, it is reflexive, symmetric, and transitive.
Question 1(v): Determine whether each of the following relations is reflexive, symmetric, and Transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R={(x,y):x and y work at the same place}
Answer:
R={(x,y):x and y work at the same place}
(x,x)∈R ,so it is reflexive
(x,y)∈R means x and y work at the same place.
y and x work at the same place, i.e. (y,x)∈R, so it is symmetric.
(x,y),(y,z)∈R means x and y work at the same place, and also y and z work at the same place. It states that x and z work at the same place, i.e. (x,z)∈R. So, it is transitive.
Hence, it is reflexive, symmetric, and transitive.
Question 1(v): Determine whether each of the following relations is reflexive, symmetric, and Transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(b) R={(x,y):x and y live in the same locality}
Answer:
R={(x,y):x and y live in the same locality}
(x,x)∈R as x and x is same human being.So, it is reflexive.
(x,y)∈R means x and y live in the same locality.
It is the same as y and x live in the same locality, i.e. (y,x)∈R.
So, it is symmetric.
(x,y),(y,z)∈R means x and y live in the same locality, and y and z live in the same locality.
It implies that x and z live in the same locality, i.e. (x,z)∈R.
Hence, it is reflexive, symmetric, and transitive.
Question 1(v): Determine whether each of the following relations is reflexive, symmetric, and transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(c) R={(x,y):x is exactly 7cm taller than y}
Answer:
R={(x,y):x is exactly 7cm taller than y}
(x,y)∈R means x is exactly 7cm taller than y, but x is not taller than x, i.e. (x,x)∉R. So, it is not reflexive.
(x,y)∈R means x is exactly 7cm taller than y, but y is not taller than x, i.e. (y,x)∉R. So, it is not symmetric.
(x,y),(y,z)∈R means x is exactly 7cm taller than y, and y is exactly 7cm taller than z.
x is exactly 7cm taller than z, i.e. (x,z)∉R.
Hence, it is not reflexive, not symmetric, and not transitive.
Question 1(v): Determine whether each of the following relations is reflexive, symmetric, and Transitive:
(v). Relation R in the set A of human beings in a town at a particular time given by
Answer:
R={(x,y):x is wife of y}
(x,y)∈R means x is the wife of y, but x is not the wife of x, i.e. (x,x)∉R.
So, it is not reflexive.
(x,y)∈R means x is wife of y but y is not wife of x i.e. (y,x)∉R .
So, it is not symmetric.
Let (x,y),(y,z)∈R mean x is the wife of y and y is the wife of z.
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 is reflexive, symmetric, and transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(e) R={(x,y):x is father of y}
Answer:
R={(x,y):x is father of y}
(x,y)∈R means x is the father of y, then x cannot be the father of x, i.e. (x,x)∉R. So, it is not reflexive.
(x,.∈R means x is the father of y, then y cannot be the father of x, i.e. (y,x)∉R. So, it is not symmetric.
Let (x,y),(y,z)∈R mean x is the father of y and y is the father of z, then x cannot be the father of z, i.e. (x,z)∉R.
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.
Answer:
R is defined in the set {1,2,3,4,5,6}
R={(a,b):b=a+1}
R={(1,2),(2,3),(3,4),(4,5),(5,6)}
Since, {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}∉R so it is not reflexive.
{(1,2),(2,3),(3,4),(4,5),(5,6)}∈R but {(2,1),(3,2),(4,3),(5,4),(6,5)}∉R
So, it is not symmetric
{(1,2),(2,3),(3,4),(4,5),(5,6)}∈R but {(1,3),(2,4),(3,5),(4,6)}∉R
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 R={(a,b): a≤b}, is reflexive and transitive but not symmetric.
Answer:
R={(a,b):a≤b}
As (a,a)∈R so I t is reflexive.
Now, we take an example
(2,3)∈R as 2<3
But (3,2)∉R because 2≮3.
So, it is not symmetric.
Now, if we take (2,3)∈Rand(3,4)∈R
Then, (2,4) because 2<4
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 is defined by R={(a,b):
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.
Answer:
Let A= {1,2,3}
R={(1,2),(2,1)}
We can see (1,1),(2,2),(3,3)∉R so it is not reflexive.
As (1,2)∈Rand(2,1)∈R, so it is symmetric.
(1,2)∈Rand(2,1)∈R
But (1,1)∉R, so it is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.
Answer:
A = all the books in the 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 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:
A={1,2,3,4,5}
R={(a,b):|a−b| is even}
R={(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(2,4),(3,5),(3,1),(5,1),(4,2),(5,3)}
Let there be a∈A, then (a, a)∈R as | a−a|=0, which is an even number. Hence, it is reflexive
Let (a,b)∈R where a,b∈A then (b,a)∈R as |a−b|=|b−a|
Hence, it is symmetric
Now, let (a,b)∈Rand(b,c)∈R
|a−b| and |b−c| are even number i.e. (a−b)and(b−c) are even
then, (a−c)=(a−b)+(b−c) is even (sum of even integer is even)
So, (a,c)∈R. Hence, it is transitive.
Thus, it is reflexive, symmetric, and transitive, i.e. it is an equivalence relation.
The elements of {1,3,5} are related to each other because the difference of odd numbers gives even numbers er and in this set, all numbers are odd.
The elements of {2,4} are related to each other because the difference of even numbers is an even number er and in this set, all numbers are even.
The element of {1,3,5} is not related to {2,4} because the difference of odd and even numbers is even.
Question 9(i): Show that each of the relation R in the set A={x∈Z:0≤x≤12}, given by (i) R={(a,b):|a−b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1 in each case.
Answer:
A={x∈Z:0≤x≤12}
A={0,1,2,3,4,5,6,7,8,9,10,11,12}
R={(a,b):|a−b| is a multiple of 4}
For a∈A, (a, a)∈ as |a−a|=0, which is a multiple of 4.
Hence, it is reflexive.
Let (a,b)∈R, i.e. |a−b| be a multiple of 4.
then |b−a| is also multiple of 4 because |a−b| = |b−a| i.e. (b,a)∈R
Hence, it is symmetric.
Let (a,b)∈R, i.e. |a−b| be a multiple of 4 and (b,c)∈R, i.e. |b−c| be a multiple of 4.
(a−b) is a multiple of 4, and (b−c) is a multiple of 4
(a−c)=(a−b)+(b−c) is multiple of 4
|a−c| is a multiple of 4, i.e. (a,c)∈R
Hence, it is transitive.
Thus, it is reflexive, symmetric, and train, positive, i.e. it is an equivalence relation.
The set of all elements related to 1 is {1,5,9}
|1−1|=0 is a multiple of 4.
|5−1|=4 is a multiple of 4.
|9−1|=8 is a multiple of 4.
Question 9(ii): Show that each of the relations R in the set A={x∈Z:0≤x≤12}, given by (ii) R={(a,b):a=b}, is an equivalence relation. Find the set of all elements related to 1 in each case.
Answer:
A={x∈Z:0≤x≤12}
A={0,1,2,3,4,5,6,7,8,9,10,11,12}
R={(a,b):a=b}
For a∈A , (a,a)∈R as a=a
Hence, it is reflexive.
Let (a,b)∈R, i.e. a=b
a=b ⇒ b=a i.e. (b,a)∈R
Hence, it is symmetric.
Let, (a,b)∈R i.e. a=b and (b,c)∈R i.e. b=c
∴ a=b=c
a=c i.e. (a,c)∈R
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
A={1,2,3}
R={(1,2),(2,1)}
(1,1),(2,2),(3,3)∉R so it is not reflexive.
(1,2)∈R and (2,1)∈R, so it is symmetric.
(1,2)∈Rand(2,1)∈R but (1,1)∉R so it is not transitive.
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
R={(x,y):x>y}
Now for x∈R , (x,x)∉R so it is not reflexive.
Let (x,y)∈R i.e. x>y
Then y>x is not possible, i.e. (y,x)∉R. So it is not symmetric.
Let (x,y)∈R i.e. x>y and (y,z)∈R i.e. y>z
we can write this as x>y>z
Hence, x>z, i.e. (x,z)∈R. So it's transitive.
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
A={1,2,3}
Define a relation R on A as
R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}
If x∈A , (x,x)∈R i.e. {(1,1),(2,2),(3,3)}∈R . So it is reflexive.
If x,y∈A , (x,y)∈R and (y,x)∈R i.e. {(1,2),(2,1),(2,3),(3,2)}∈R . So it is symmetric.
(x,y)∈R and (y,z)∈R i.e. (1,2)∈R . and (2,3)∈R
But (1,3)∉R, so it is not transitive.
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
R={(a,b):a≤b}
(a, a)∈R because a=a
Let (a,b)∈R, i.e. a≤b
But (b, a)∉R, i.e. b⪇a
So it is not symmetric.
Let (a,b)∈R i.e. a≤b and (b,c)∈R i.e. b≤c
This can be written as a≤b≤c i.e. a≤c implies (a,c)∈R
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
A={1,2}
R={(1,2),(2,1),(2,2)}
(1,1)∉R So R is not reflexive.
We can see (1,2)∈R and (2,1)∈R
So it is symmetric.
Let (1,2)∈R and (2,1)∈R
Also (2,2)∈R
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 the same point P from another origin, i.e.
Let
This is the same as the distance of point Q from the origin, same as the distance of point P from the origin, i.e.
Let
i.e. distance of the point P from the origin is the same as the distance of the point
We can say that the distance of points
Hence,
The set of all points related to a point
In other words, we can say there be a point
Hence, this set of points forms a circle with the centre as the origin, and this circle passes through point
Answer:
All triangles are similar to themselves, so it is reflexive.
Let,
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
Now, we see the ratio of the sides of triangles T1 and T3 as shown
i.e. ratios of sides of T1 and T3 are equal. Hence,
Answer:
The same polygon has the same number of sides with itself,i.e.
Let,
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
The elements in A related to the right angle triangle
Hence, the set of all elements in A related to the right-angle triangle T is a set of all triangles.
Answer:
All lines are parallel to themselves, so it is reflexive.
Let,
Hence, it is symmetric.
Let,
Hence, it is transitive,
Thus,
The set of all lines related to the line
Here, Slope
It is known that the slope of parallel lines is equal.
Lines parallel to this
Hence, the set of all parallel lines to
(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}
R={(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}
For every a∈A, there is (a, a)∈R.
∴ R is reflexive.
Given, (1,2)∈R but (2,1)∉R
∴ R is not symmetric.
For a,b,c∈A there are (a,b)∈Rand(b,c)∈R ⇒ (a,c)∈R
∴ R is transitive.
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 R={(a,b):a=b−2,b>6}. Choose the correct answer.
Answer:
R={(a,b):a=b−2,b>6}
(A) Since b<6, so (2,4)∉R
(B) Since 3≠8−2, so (3,8)∉R
(C) Since, 8>6 and 6=8−2 so (6,8)∈R
(D) Since 8≠7−2, so (8,7)∉R
The correct answer is option C.
NCERT Relations and Functions Class 11 Solutions: Exercise: 1.2 Page Number: 10-11 Total Questions: 12 |
Answer:
Given,
One - One :
Onto:
We have
Hence, the function is one-one and onto.
If the domain
For
Hence, function g is one-one but not onto.
Question 2 (i): Check the injectivity and surjectivity of the following functions:
Answer:
One-one:
For
Hence,
Question 2 (ii): Check the injectivity and surjectivity of the following functions:
Answer:
One-one:
For
For
Hence,
Question 2 (iii): Check the injectivity and surjectivity of the following functions:
Answer:
One-one:
For
For
Hence,
Question 2 (iv): Check the injectivity and surjectivity of the following functions:
Answer:
One-one:
For
Hence,
Question 2 (v): Check the injectivity and surjectivity of the following functions:
Answer:
One-one:
For
For
Hence,
Answer:
f:R⟶R
f(x)=[x]
One-one:
For 1.5,1.7∈R then f(1.5)=[1.5]=1 and f(1.7)=[1.7]=1
but 1.5≠1.7
∴ f is not one-one, i.e. not injective.
For 0.6∈R there is no x in R such that f(x)=[0.6]
∴ f is not onto i.e. not surjective.
Hence, f is not injective but not surjective.
Answer:
f: R→R
f(x)=|x|
f(x)=|x|=xifx≥0and−xifx<0
One-one:
For −1,1∈R then f(−1)=|−1|=1
f(1)=|1|=1
−1≠1
∴ f is not one-one, i.e. not injective.
For −2∈R,
We know f(x)=|x| is always positive there is no x in R such that f(x)=|x|=−2
∴ f is not onto i.e. not surjective.
Hence, f(x)=|x| is neither one-one nor onto.
Question 5: Show that the Signum Function f: R→R, given by f(x)={1 if x>0 0 if x=0 −1 if x<0, is neither one-one nor onto.
Answer:
As we can see
So it is not one-one.
Now,
So it is not onto.
Hence, the signum function is neither one-one nor onto.
Question 6: Let A={1,2,3}, B={4,5,6,7} and let f={(1,4),(2,5),(3,6)} be a function from A to B. Show that f is one-one.
Answer:
A={1,2,3}
B={4,5,6,7}
f={(1,4),(2,5),(3,6)}
f: A→B
∴ f(1)=4,f(2)=5,f(3)=6
Every element of A has a distant value in f.
Hence, it is one-one.
Question 7 (i): In each of the following cases, state whether the function is one-one, onto, or bijective. Justify your answer.
(i) f:R→R defined by f(x)=3−4x
Answer:
f: R→R
f(x)=3−4x
Let there be (a,b)∈R such that f(a)=f(b)
3−4a=3−4b
−4a=−4b
a=b
∴ f is one-one.
Let there be y∈, y=3−4x
x=(3−y)4
f(x)=3−4x
Putting value of x, f(3−y4)=3−4(3−y4)
f(3−y4)=y
∴ f is onto.
f is both one-one and onto; hence, f is bijective.
Question 7 (ii): In each of the following cases, state whether the function is one-one, onto, or bijective. Justify your answer.
(ii) f:R→R defined by f(x)=1+x^2
Answer:
Let there be
For
Let there be
There does not exists any x in domain R such that
Hence, f is neither one-one nor onto.
Question 8: Let A and B be sets. Show that f: A×B→B×A such that f(a,b)=(b, a) is a bijective function.
Answer:
f: A×B→B×A
f(a,b)=(b,a)
Let (a1,b1),(a2,b2)∈A×B
such that f(a1,b1)=f(a2,b2)
(b1,a1)=(b2,a2)
⇒ b1=b2 and a1=a2
⇒ (a1,b1)=(a2,b2)
∴ f is one- one
Let (b, a)∈B×A
then there exists (a,b)∈A×B such that f(a,b)=(b,a)
∴ f is onto.
Hence, it is bijective.
Question 9: Let
Answer:
Here we can observe,
As we can see
Let,
case1
For
then there is
case2 n be odd
For
then there is
Hence, the function
Question 10: Let
Answer:
Let
Let,
For any
Hence, the function is one-one and onto.
Question 11: Let f: R→R be defined as f(x)=x^4. Choose the correct answer.
(D) f is neither one-one nor onto.
Answer:
One-one:
For
example :
For
Hence,
Hence, option D is correct.
Question 12: Let f: R→R be defined as f(x)=3x. Choose the correct answer.
(D) f is neither one-one nor onto.
Answer:
f: R→R
f(x)=3x
One - One :
Let (x,y)∈R
f(x)=f(y)
3x=3y
x=y
∴ f is one-one.
Onto:
If we have y∈R, then there exists x=y/3∈R such that
f(y/3)=3×y/3=y
∴f is onto.
Hence, the function is one-one and onto.
The correct answer is A.
NCERT Relation and Functions Class 12 Solutions: Exercise: Miscellaneous Exercise Page Number: 15-16 Total Questions: 7 |
Question 1: Show that the function f:R→{x∈R:−1<x<1} defined by f(x)=x/(1+|x|) x∈R is one one and onto function.
Answer:
The function
One-one:
Let
It is observed that if x is positive and y is negative.
Since x is positive and y is negative.
Thus, the case of x being positive and y being negative is removed.
The same happens in the case of y is positive and x is negative, so this case is also removed.
When x and y both are positive:
When
Onto:
Let
If y is negative, then
If y is positive, then
Thus,
Hence,
Question 2: Show that the function f: R→R given by f(x)=x^3 is injective.
Answer:
f: R→R
f(x)=x^3
One-one:
Let f(x)=f(y)x,y∈R
x^3=y^3
We need to prove x=y. So,
Let x≠y, then their cubes will not be equal, i.e. x^3≠y^3.
It will contradict the given condition of cubes being equal.
Hence, x=y, and it is one-one, which means it is injective
Answer:
Given a nonempty set X, consider
Since every set is a subset of itself, ARA for all
Let
This is not the same as
If
then we cannot say that
If
this implies
Thus, R is not an equivalence relation because it is not symmetric.
Question 4: Find the number of all onto functions from the set {1,2,3,...,n} to itself.
Answer:
The number of all onto functions from the set {1,2,3,...,n} to itself is the permutations of n symbols 1,2,3,4,5...............n.
Hence, permutations on n symbols 1,2,3,4,5...............n = n
Thus, the total number of all onto maps from the set {1,2,3,...,n} to itself is the same as permutations on n symbols 1,2,3,4,5...............n, which is n.
Question 5: Let
Answer:
Given :
It can be observed that.
Hence,
Answer:
A={1,2,3}
The smallest relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive, are given by
R={(1,1),(2,2),(3,3),(1,2),(1,3),(2,1),(3,1)}
(1,1),(2,2),(3,3)∈R , so relation R is reflexive.
(1,2),(2,1)∈R and (1,3),(3,1)∈R , so relation R is symmetric.
(2,1),(1,3)∈R but (2,3)∉R , so realation R is not transitive.
Now, if we add any two pairs (2,3) and (3,2) to relation R, then relation R will become transitive.
Hence, the total number of the desired relation is one.
Thus, option A is correct.
Question 7: Let A={1,2,3}. The number of equivalence relations containing (1,2) is
(A) 1
(B) 2
(C) 3
(D) 4
Answer:
A={1,2,3}
The number of equivalence relations containing (1,2) is given by
R={(1,1),(2,2),(3,3),(1,2),(2,1)}
We are left with four pairs (2,3), (3,2), (1,3),(3,1).
(1,1),(2,2),(3,3)∈R , so relation R is reflexive.
(1,2),(2,1)∈R and (2,3),(3,2)∉R , so relation R is not symmetric.
(1,3),(3,1)∉R , so realation R is not transitive.
Hence, the equivalence relation is bigger than R is the universal relation.
Thus, the total number of equivalence relations containing (1,2) is two.
Thus, option B is correct.
Question: Let an operation
Find (i) whether
Solution:
(i) As
ie
Hence,
(ii) As
so
Hence,
Also read,
Relations and Functions can feel like a puzzle of definitions, diagrams, and properties—but with the right approach, every question starts to make perfect sense.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
Also read,
Students can check the following links for more in-depth learning.
Students can check the following links for more in-depth learning.
NCERT Class 12 Maths Chapter 1 "Relations and Functions" covers essential standards of set principles and mappings. Key subjects encompass:
Types of Relations – Reflexive, symmetric, transitive, and equivalent members of the family.
Types of Functions – One-one (injective), onto (surjective), and bijective functions.
Composition of Functions – Understanding how functions integrate.
The inverse of a Function – Conditions for the lifestyles of an inverse.
Binary Operations – Definition consisting of commutativity and associativity.
In NCERT Class 12 Maths, one-one (injective) and onto (surjective) functions are two important types of functions:
- One-One Function (Injective): A function is one-one if different inputs give different outputs. This means no two elements in the domain map to the same element in the codomain. Example: f(x)=2x is one-one.
- Onto Function (Surjective): A function is onto if every element in the codomain has at least one pre-image in the domain. This means the function covers the entire codomain. Example: f(x)=x3 is onto for real numbers.
A function can be both one-one and onto (bijective).
In NCERT Class 12 Maths Chapter 1, relations and functions are different concepts:
- Relation: A relation connects elements of one set to another. It is simply a pairing of elements but may not follow specific rules. Example: If Set A={1,2,3} and Set B={4,5}, a relation can be {(1,4), (2,5)}.
- Function: A function is a special type of relation where each input has exactly one output. Example: f(x)=x+2 maps every x to a unique value.
Thus, every function is a relation, but not every relation is a function.
In NCERT Class 12 Maths Chapter 1, the composition of functions means applying one function to the result of another. If we have two functions f(x) and g(x), their composition is written as (f o g)(x), which means f(g(x)).
Steps to find composition:
1. First, find g(x) (solve for x in g ).
2. Substitute g(x) into f(x).
3. Simplify the expression.
For example, if f(x)=x2 and g(x)=x+1, then:
(f o g)(x)=f(g(x))=f(x+1)=(x+1)2
In NCERT Class 12 Maths Chapter 1, the domain and range of a function describe its input and output values:
- Domain: The set of all possible input values (x) for which the function is defined. Example: For f(x)=1/x, the domain is all real numbers except x=0.
- Range: The set of all possible output values f(x). Example: For f(x)=x2, the range is [0, infinity) because squares are always non-negative.
Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
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If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
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