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Relations and Functions Class 11 Notes Maths Chapter 2

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 Page 1


 
KEY POINTS
? Cartesian Product of two non-empty sets A and B is given by,A 
× B = { (a, b) : a ? A, b ? B} 
? If (a, b) = (x, y), then a = x and b = y 
? Relation R from a non-empty set A to a non-empty set B is a 
subset of A×B. 
? Domain of R = {a : (a, b) ? R} 
? Range of R = { b : (a, b) ? R} 
? Co-domain of R = Set B 
? Range ? Co-domain 
? If n(A) = p, n(B) = q then n(A×B) = pq and number of relations = 
2
pq
 
? Image : If the element x of A corresponds to y ? B under the 
function f, then we say that y is image of x under ‘f ’ 
? f (x) = y 
? If f (x) = y, then x is preimage of y. 
? A relation f from a set A to a set B is said to be a function if 
every element of set A has one and only one image in set B. 
Page 2


 
KEY POINTS
? Cartesian Product of two non-empty sets A and B is given by,A 
× B = { (a, b) : a ? A, b ? B} 
? If (a, b) = (x, y), then a = x and b = y 
? Relation R from a non-empty set A to a non-empty set B is a 
subset of A×B. 
? Domain of R = {a : (a, b) ? R} 
? Range of R = { b : (a, b) ? R} 
? Co-domain of R = Set B 
? Range ? Co-domain 
? If n(A) = p, n(B) = q then n(A×B) = pq and number of relations = 
2
pq
 
? Image : If the element x of A corresponds to y ? B under the 
function f, then we say that y is image of x under ‘f ’ 
? f (x) = y 
? If f (x) = y, then x is preimage of y. 
? A relation f from a set A to a set B is said to be a function if 
every element of set A has one and only one image in set B. 
 
? Df = {x : f(x) is defined} Rf = {f(x) : x ? Df} 
? Let A and B be two non-empty finite sets such that n(A) = p and 
n(B) = q then number of functions from A to B = q
p
. 
? Identity function, f : R ? R; f(x) = x ? x ? R, where R is the set 
of realnumbers. 
Df = R Rf = R 
? Constant function, f : R ? R; f(x) = c ? x ? R where c is a 
constant 
Df = R Rf = {c} 
? Modulus function, f : R ? R; f(x) = |x| ? x ? R 
Df = R 
Rf = R
+
? { 0} = { x : x ? R: x ? 0} 
O X X´ 
Y 
Y´ 
f 
( 
x 
) 
 
= 
 
x
Page 3


 
KEY POINTS
? Cartesian Product of two non-empty sets A and B is given by,A 
× B = { (a, b) : a ? A, b ? B} 
? If (a, b) = (x, y), then a = x and b = y 
? Relation R from a non-empty set A to a non-empty set B is a 
subset of A×B. 
? Domain of R = {a : (a, b) ? R} 
? Range of R = { b : (a, b) ? R} 
? Co-domain of R = Set B 
? Range ? Co-domain 
? If n(A) = p, n(B) = q then n(A×B) = pq and number of relations = 
2
pq
 
? Image : If the element x of A corresponds to y ? B under the 
function f, then we say that y is image of x under ‘f ’ 
? f (x) = y 
? If f (x) = y, then x is preimage of y. 
? A relation f from a set A to a set B is said to be a function if 
every element of set A has one and only one image in set B. 
 
? Df = {x : f(x) is defined} Rf = {f(x) : x ? Df} 
? Let A and B be two non-empty finite sets such that n(A) = p and 
n(B) = q then number of functions from A to B = q
p
. 
? Identity function, f : R ? R; f(x) = x ? x ? R, where R is the set 
of realnumbers. 
Df = R Rf = R 
? Constant function, f : R ? R; f(x) = c ? x ? R where c is a 
constant 
Df = R Rf = {c} 
? Modulus function, f : R ? R; f(x) = |x| ? x ? R 
Df = R 
Rf = R
+
? { 0} = { x : x ? R: x ? 0} 
O X X´ 
Y 
Y´ 
f 
( 
x 
) 
 
= 
 
x
 
? ?Signum function 
1,if x >0
x
, x 0
f : R R ; f (x) = 0,if x = 0 and f (x) = 
x
0,x = 0 — 1,if x < 0
?
?
? ? ?
?
? ?
? ?
?
?
Then  
Df = R 
and Rf = {–1,0,1} 
? Greatest Integer functionf : R ? R; f(x) = [x], x ? R assumes the 
value of the greatest integer, less than or equal to x 
Df = R Rf = Z 
O 
X X´ 
Y 
Y´ 
1 
y  = 1 
y = – 1 
– 1
O 
X X´ 
Y 
Y´ 
Page 4


 
KEY POINTS
? Cartesian Product of two non-empty sets A and B is given by,A 
× B = { (a, b) : a ? A, b ? B} 
? If (a, b) = (x, y), then a = x and b = y 
? Relation R from a non-empty set A to a non-empty set B is a 
subset of A×B. 
? Domain of R = {a : (a, b) ? R} 
? Range of R = { b : (a, b) ? R} 
? Co-domain of R = Set B 
? Range ? Co-domain 
? If n(A) = p, n(B) = q then n(A×B) = pq and number of relations = 
2
pq
 
? Image : If the element x of A corresponds to y ? B under the 
function f, then we say that y is image of x under ‘f ’ 
? f (x) = y 
? If f (x) = y, then x is preimage of y. 
? A relation f from a set A to a set B is said to be a function if 
every element of set A has one and only one image in set B. 
 
? Df = {x : f(x) is defined} Rf = {f(x) : x ? Df} 
? Let A and B be two non-empty finite sets such that n(A) = p and 
n(B) = q then number of functions from A to B = q
p
. 
? Identity function, f : R ? R; f(x) = x ? x ? R, where R is the set 
of realnumbers. 
Df = R Rf = R 
? Constant function, f : R ? R; f(x) = c ? x ? R where c is a 
constant 
Df = R Rf = {c} 
? Modulus function, f : R ? R; f(x) = |x| ? x ? R 
Df = R 
Rf = R
+
? { 0} = { x : x ? R: x ? 0} 
O X X´ 
Y 
Y´ 
f 
( 
x 
) 
 
= 
 
x
 
? ?Signum function 
1,if x >0
x
, x 0
f : R R ; f (x) = 0,if x = 0 and f (x) = 
x
0,x = 0 — 1,if x < 0
?
?
? ? ?
?
? ?
? ?
?
?
Then  
Df = R 
and Rf = {–1,0,1} 
? Greatest Integer functionf : R ? R; f(x) = [x], x ? R assumes the 
value of the greatest integer, less than or equal to x 
Df = R Rf = Z 
O 
X X´ 
Y 
Y´ 
1 
y  = 1 
y = – 1 
– 1
O 
X X´ 
Y 
Y´ 
 
? f : R ? R, f(x) = x
2
 
Df = R Rf = [0, ? ? 
? f : R ? R, f(x) = x
3
 
Df = R Rf = R 
? Exponential function, f : R ? R ; f(x) = a
x
, a > 0, a ? 1 
X´ 
O 
X 
Y 
Y´ 
O 
X X´ 
Y 
Y´ 
O 
X X´ 
Y 
Y´
2 
1 
–2 –1 1 2 3 4 
–1
–2
3 
Page 5


 
KEY POINTS
? Cartesian Product of two non-empty sets A and B is given by,A 
× B = { (a, b) : a ? A, b ? B} 
? If (a, b) = (x, y), then a = x and b = y 
? Relation R from a non-empty set A to a non-empty set B is a 
subset of A×B. 
? Domain of R = {a : (a, b) ? R} 
? Range of R = { b : (a, b) ? R} 
? Co-domain of R = Set B 
? Range ? Co-domain 
? If n(A) = p, n(B) = q then n(A×B) = pq and number of relations = 
2
pq
 
? Image : If the element x of A corresponds to y ? B under the 
function f, then we say that y is image of x under ‘f ’ 
? f (x) = y 
? If f (x) = y, then x is preimage of y. 
? A relation f from a set A to a set B is said to be a function if 
every element of set A has one and only one image in set B. 
 
? Df = {x : f(x) is defined} Rf = {f(x) : x ? Df} 
? Let A and B be two non-empty finite sets such that n(A) = p and 
n(B) = q then number of functions from A to B = q
p
. 
? Identity function, f : R ? R; f(x) = x ? x ? R, where R is the set 
of realnumbers. 
Df = R Rf = R 
? Constant function, f : R ? R; f(x) = c ? x ? R where c is a 
constant 
Df = R Rf = {c} 
? Modulus function, f : R ? R; f(x) = |x| ? x ? R 
Df = R 
Rf = R
+
? { 0} = { x : x ? R: x ? 0} 
O X X´ 
Y 
Y´ 
f 
( 
x 
) 
 
= 
 
x
 
? ?Signum function 
1,if x >0
x
, x 0
f : R R ; f (x) = 0,if x = 0 and f (x) = 
x
0,x = 0 — 1,if x < 0
?
?
? ? ?
?
? ?
? ?
?
?
Then  
Df = R 
and Rf = {–1,0,1} 
? Greatest Integer functionf : R ? R; f(x) = [x], x ? R assumes the 
value of the greatest integer, less than or equal to x 
Df = R Rf = Z 
O 
X X´ 
Y 
Y´ 
1 
y  = 1 
y = – 1 
– 1
O 
X X´ 
Y 
Y´ 
 
? f : R ? R, f(x) = x
2
 
Df = R Rf = [0, ? ? 
? f : R ? R, f(x) = x
3
 
Df = R Rf = R 
? Exponential function, f : R ? R ; f(x) = a
x
, a > 0, a ? 1 
X´ 
O 
X 
Y 
Y´ 
O 
X X´ 
Y 
Y´ 
O 
X X´ 
Y 
Y´
2 
1 
–2 –1 1 2 3 4 
–1
–2
3 
 
Df = R Rf= (0, ?) 
0 < a < 1 a > 1 
? Natural exponential function, f(x) = e
x
 
1 1 1
1 ... , 2 3
1! 2! 3!
e e ? ? ? ? ? ? ? ?
? Logarithmic functions, f : (0, ?) ? R ; f(x)lo g
a
x, a > 0, a ? 1
Df = (0, ?) 
R
f = R 
? Natural logarithrnic function f(x) = logex or log x. 
? Let f : X ? R and g : X ? R be any two real functions where x ? 
R then 
(f ± g) (x) = f(x) ± g(x) ? x ? X 
(fg) (x) = f(x) g(x) ? x ? X 
X 
Y 
Y’ 
X’ 
X 
Y 
Y’ 
X
’ 
(0 ,  1) 
(0 ,  1) 
O O 
?
?
? ? 0
f ? x
f
X provided g ? x
g ? x
? ?
?
g
?
? x ? ? ? x ?
? ?
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FAQs on Relations and Functions Class 11 Notes Maths Chapter 2

1. What is a relation in mathematics?
Ans. A relation in mathematics is a set of ordered pairs that establish a connection or association between two sets of elements. It can be seen as a way of relating one set to another.
2. What are the different types of relations?
Ans. There are several types of relations in mathematics, including: - Reflexive relations: Every element in a set is related to itself. - Symmetric relations: If A is related to B, then B is also related to A. - Transitive relations: If A is related to B and B is related to C, then A is related to C. - Equivalence relations: A relation that is reflexive, symmetric, and transitive.
3. How do you determine if a relation is a function?
Ans. To determine if a relation is a function, we need to check if each input value (x) is associated with only one output value (y). In other words, for every x-value, there should be only one corresponding y-value. If this condition is satisfied, the relation is a function; otherwise, it is not.
4. What is the difference between a relation and a function?
Ans. A relation is a set of ordered pairs that establish a connection between two sets, while a function is a special type of relation where each input value is associated with only one output value. In other words, every x-value in a function has a unique y-value, whereas this is not necessary in a general relation.
5. How can I graph a relation or function?
Ans. To graph a relation or function, you can plot the ordered pairs on a coordinate plane. For each ordered pair (x, y), you can locate the point (x, y) on the plane. By connecting all the points, you can visualize the relation or function. If the relation is a function, the graph should pass the vertical line test, meaning that no vertical line intersects the graph at more than one point.
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