Relations and Functions Class 11

Ex2.1,1 teachoo
x 2 51 a
If G +1y -2) = G5) , find the values of x and y.
x 2 5 1
G+ry -3)=G3)
Since the ordered pairs are equal, corresponding elements are equal.
Hence
201
X4425 V-353
3 +1= 3 3° 3
x 5 1,2
37a t Y=3"3
x_2 3
3.3 Y"3
x=2 y=1

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Ex 2.1, 2
If the set A has 3 elements and the set B = {3, 4, 5}, then find the
number of elements in (A x B)?
Number of elements in set A = 3
Number of elements inset B=3 (Given set B = {3, 4, 5})
Number of elements in set A x B
= Number of elements in set A x Number of elements in set B
=3x3
=9
Thus, the number of elements in (A x B) is 9.

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Ex 2.1, 3
If G = {7, 8} and H = {5, 4, 2}, find G x Hand HxG.
G ={7, 8}
and H = {5, 4, 2}
GxH={7, 8}x {5, 4, 2}
Gx H = {(7, 5),(7, 4), (7, 2),(8, 5),(8, 4), (8, 2) }
Hx G={5, 4, 2} x {7, 8}
Hx G={(5, 7),(5, 8), (4, 7),(4, 8),(2, 7), (2, 8) }

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Ex 2.1, 4
State whether each of the following statements are true or false. If
the statement is false, rewrite the given statement correctly.
{i) If P= {m, n} and Q={1n, m}, then P xQ = {(m, n)},{n, m)}.
P={m,n}
Q={n, m}
‘Px Q= {m, n} x {n, m}

= {(m, n), (m, m),

(n, n), (n, m)}

Hence. Px Q # {(m,n),{n, m)}.
Hence, the given statement is False

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ

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Ex 2.1,5
If A={-1, 1}, find AXAXA.
AxAxA= 13x11 xC1L 1
= {(-1, -1, -1), (-1, -1, 1),

(-1,1,-1), (-1,1,1),

(1,-1,-1), (1,-1,1),

(1,1,-1), (1,1, 1}

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Ex 2.1, 6
If Ax B= {{a, x), (a, y), (b, x), (b, y)}. Find A and B.
Given A x B = {(a, x), (a, y), (b, x), (b, y)}
Ais the set of all first elements
i.e. A={a,b} (Since first element contains only a and b)
and
B is the set of all second elements.
B= {x, y} (Since second element contains only x and y)

Ex 2.1, 7 teachoo.com
Let A = {1, 2}, B={1, 2, 3, 4}, C={5, 6}and D ={5, 6, 7, 8}. Verify that
(i) Ax(BOC)=(AxB)N(AxC
L.H.S R.H.S
Ax(BNC) (Ax B) 9 (Ax C)
AxB={1,2 1,2,3,4
BAC xB={1,2}x{ }
= {(1, 1), (1, 2), (1, 3), (1, 4),
=(1,2,3,4} 05, 6} {(1, 1), (1, 2), (1, 3), (1 4)
(2, 1), (2, 2), (2, 3), (2, 4)}
=
AxC=({1, 2} x {5, 6}
Ax{B OC) ={ (1,5), (1, 6),
={L xo (2, 5), (2, 6)}
=o
(Ax B) 9 (AxC)
=@

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Ex 2.1, 8
Let A= {1, 2} and B = {3, 4}. Write A x B. How many subsets
will A x B have? List them.
A= {1,2} & B= {3, 4}
Ax B= {(1, 3), (1, 4), (2, 3), (2, 4)}
No of elements of Ax B=n=4
Number of subsets of Ax B= 2"

=24

=2x2x2x?

=16
Therefore, the set A x B has 24 = 16 subsets.

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Ex 2.1,9
Let A and B be two sets such that n(A) = 3 and n (B) = 2. If
(x, 1}, (y, 2), (z, 1) are in A x B, find A and B, where x, y and z
are distinct elements.
Ax B contains (x, 1), (y, 2), (z, 1)
Ais the set of all first elements
ie. A= {x, y, Z} (Since first element contains x, y and z)
and
B is the set of all second elements.
B = {1, 2} (Since second element contains only 1 and 2)

teackoo.com
Ex 2.1, 10
The Cartesian product A x A has 9 elements among which
are found (—1, 0} and (0, 1). Find the set A and the
remaining elements of A x A.
Since A x A has 9 elements
So, A would have 3 elements (As 3 x 3 = 9)
Let A = {a, b, c}
Now, A x A = {a, b, c} x {a, b, c}
= { (a, a), (a, b), (a, ¢),
(b, a), (b, b), (b, c),
{c, a), (c, b), (c, c}}

Ex 2.2, 1 teachoo
Let A = {1, 2, 3, ..., 14}. Define a relation R from A to A by R = {{x, y):
3x—y = 0, where x, y € A}. Write down its domain, codomain and
range.
= 3x x,yEA
It is given that u y
1 3x1 =3 Yes
3x-y=0
2 3x2 =6 Yes
3x=y 3 3x3 =9 Yes
y= 3x 4 3x4 =12 Yes
S 3xS =15 No
6 3x6 =18 No
Finding Relation R 7 3x7 =21 No
First, we check which values of x, y is in set A
If both x and y are in set A, then (x, y) ER

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Ex 2.2, 2
Define a relation R on the set N of natural numbers by R = {(x, y): y =x
+5,x is a natural number less than 4; x, y € N}. Depict this
relationship using roster form. Write down the domain and the range.
Given y=x+5
Also, x is a natural number less than 4
Hence, x is 1, 2, and 3.
ra

1 14+5=6

2 2+5 =7

3 3+5 =8
Hence, R = {(1, 6), (2, 7), (3, 8)}

Ex 2.2, 3 teackoo.com
A ={1, 2, 3, 5}and B = {4, 6, 9}.
Define a relation R from A to B by R = {(x, y): the difference between
x and y is odd; x € A, y € B}. Write R in roster form.
Given A=({1,2, 3,5} & B= {4, 6, 9}
Here, xEA&y EB
It is given that difference between x and y is odd
Hence, various combinations possible are
Value of y | Difference (y—x) | Whether difference odd
1 4 4-1=3 Yes
1 6 6-1=5 Yes
1 9 9-1=8 No
2 4 4-2=2 No
2 6 6-2=4 No

2.2, 4 teachoo.com
The given figure shows a relationship between the sets P and Q.
write this relation P °
{i) in set-builder form
|
Note that (4
5-3=2
6-4=2
7-5=2
So, difference of elements is 2
Let the elements of set P can be denoted by x i.e. x € P
& the elements of set Q can be denoted by yi.e. yEQ

Ex 2.2, 5 teachoo.com
Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a, b
EA, bis exactly divisible by a}.
(i) Write R in roster form
A={1, 2, 3, 4, 6}
a,bEA
Also, b is exactly divisible by a
Pasa eet) [are
1 1 : =1 Yes
1 2 222 Yes
1
1 3 : =3 Yes
1 4 ¢ =4 Yes
1 6 =6 Yes

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Ex 2.2, 6
Determine the domain and range of the relation R defined by
R= {(x, x +5): x € {0, 1, 2, 3, 4, 5}}.
xis between 0 to 5
Px] x+s

o oO+5 =5

1 1+5 =6

2 2+5 =7

3 3+5 =8

4 44+5 =9

5 5+5 =10
R= {(O, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}

teackoo.co:

Ex 2.2,7 "
Write the relation
R = {(x, x3): x is a prime number less than 10} in roster form.
Prime numbers = 2, 3, 5, 7, 11, 13, 17
Prime numbers less than 10 = 2, 3, 5, 7
So, value of x = 2, 3,5, 7

2 23 =2x2x2 =8

3 33 =3x3x3 =27

5 53 =5x5x5 =125

7 Po=7X7x7 =343
Hence,
R = {(2, 8), (3, 27), (5, 125), (7, 343)}

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Ex 2.2,8
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
Given A = {x y, z} & B = {1, 2}
Number of relations from A to B = 2Number of elements in Ax 8

= 2Number of elements in set A x Number of elements in set B

= 2na) x n{B)
Number of elements in set A = 3
Number of elements in set B = 2
Number of relations from A to B = 2A) * "(8)

= 932

Ex 2.2, 9 teackoo.com
Let R be the relation on Z defined by
R = {{a, b): a, b € Z, a— bis an integer}. Find the domain and range of R,
Given a, b are integers, i.e. a,b EZ
As difference of integers are integers,
a-—bis an also integer
-. Domain of R = Set of all first elements in the relation
= Values of ‘a’ which are in the relation
=Zi.e. integers

Ex 2.3,1 teackhoo
Which of the following relations are functions? Give reasons. If it is
a function, determine its domain and range.
(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
A relation is a function if
* Every element of first set has an image
¢ Every element of first set has only one image
Since both conditions are satisfied 5
So, this relation is a function. 5 —
— 1
11 I
14 i
17

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(iii) {(1, 3), (1, 5), (2, 5)}

Ex 2.3, 2 teackoo.com
Find the domain and range of the following real function:
(i) f(x) =—|x|
Here we are given a real function
Hence, both domain and range should be real numbers
Value of [x] | fo0=—tat Whether real
number
|2) =2 -2
Jap=1 -1
Jo] =o 0
J-1] =1 -1
|-2| =2 -2
Here, x can be any Here, f(x) will always be
real number negative or zero.
All these are real values

Find the domain and range of the following real function:
(ii) f(x) = √((9 −x^2))

Ex 2.3, 3 teackhoo.com
A function f is defined by f(x) = 2x — 5. Write down the values of
(i) f(0) (ii) #(7), (iii) f(-3),
Given f(x) = 2x-—5
Putting value of x in f(x)
f(0)=2x0-5 | f(7)=2x7-5 | f(-3)=2x(-3)-5
=0-5 =14-5 =-6-5
=-5 =9 =-11

teachoo.com
Ex 2.3, 4
The function ‘t’ which maps temperature in degree Celsius into
temperature in degree Fahrenheit is defined by .
9 :
t(C) = z C+ 32. Find
(i) t (0) (ii) t (28) (iii) t (-10)
Given
t(c)=2C +32
Putting C = 0,
9
t(0) = =* (0) + 32
=0+32
=32

Ex 2.3, 5 teachoo.com
Find the range of each of the following functions.
(i) f(x) = 2-3x, x ER, x>0.
Given that
x>0, (We need to make it in form 2 — 3x)
Multiplying 3 both sides
3x>O0x3
3x>0
Multiplying -1 both sides
-1x3x<-1x0
—3x<0
Adding 2 both sides
2-—3x<2+0

Find the range of each of the following functions.
(ii) f(x) = x2 + 2, x, is a real number.

Find the range of each of the following functions.
(iii) f(x) = x, x is a real number

Example 1 teackhoo
If (x + 1, y -— 2) = (3, 1), find the values of x and y.
(x+1, y-2) = (3, 1)
Since the ordered pairs are equal, corresponding elements are equal.
Hence,

x+1=3 y-2=1

x=3-1 y=142

x=2 y=3
Therefore, x = 2, y=3

Example 2 teachoo.com
If P = {a, b, c} and Q = {r}, find the sets P x Q and Qx P. Are

these two products equal?

P = {a, b, c}

and Q = {r}

Px Q={a, b, c} x {r}

Px Q={(a,r), (b, 1), (c, r)}

Qx P= {r} x {a, b, c}

QxP= {(r, a), (r, b), (r, c}}

teackoo.com

Example 3
Let A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}. Find
(i) Ax (BAC)
BN C= {3,4} n {4, 5, 6}

= {4}
Ax(BNOC)

= {1, 2, 3} x {4}

= {(1, 4), (2, 4), (3, 4)}

teackoo.co:
Example 4 oH
If P = {1, 2}, form the set P x P x P.
PxPxP= {1,2}x {1,2} {1, 2}
={(1, 1, 1), (1, 1, 2),
(1, 2, 1),(1, 2, 2),
(2, 1, 1),(2, 1, 2),
(2, 2, 1), (2, 2, 2)}

teachoo.com
Example 5
If R is the set of all real numbers, what cdo the cartesian products
Rx Rand Rx RXR represent?
Rx R= {(x, y}: x, y € R}
represents the coordinates of all the points in two dimensional
space
RxXRxR= {(x, y, Zz): x,y,z © R}
represents the coordinates of all the points in three-dimensional
space.

teackoo.com
Example 6
If A x B= {(p, q), (p, r), (m, q), (m, r)}, find A and B.
Given A x B = {(p, q), (p, r), (m, q), (m, r)}
Ais the set of all first elements
i.e. A= {p, m} (Since first element contains only p and m)
and
B is the set of all second elements.
B = {q, r} (Since second element contains only q and r)

Example 7 teackoo.com
Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by
R={(x y):y=x+1}
(i) Depict this relation using an arrow diagram.
Itis given that y=x+1
Also, x, yEA

1 1+1 =2 Yes

2 2+1=3 Yes

3 3+1 =4 Yes

4 4+1=5 Yes

5 5+1 =6 Yes

6 6+1 =7 No

teachoo.com
Example 8
The figure shows a relation between the sets P and Q.
Write this relation
P Q
(i) in set-builder form
7]
—
=
<J
(3) =9
(3-9
(2? =4
(27 =4
(5)? = 25
(-S}2= 25

Example 9
Let A = {1, 2} and B = {3, 4}. Find the number of relations from A to B.
Given A = {1,2} & B = {3,4}
Number of relations from A to B = 2Number of elements in A × B
= 2Number of elements in set A × Number of elements in set B
= 2n(A) × n(B)
Number of elements in set A = 2
Number of elements in set B = 2
Number of relations from A to B = 2n(A) × n(B)
= 22 × 2

Example 10 teackoo.com

Let N be the set of natural numbers and the relation R be

defined on N such that R = {(x, y) : y = 2x, x, y € N}. What is the

domain, codomain and range of R? Is this relation a function?
Value of x Value of y

Given that y=2x &x,yEN

Here, xis a always natural number,

So, Domain = Set of natural numbers = N

Here, y is always an even number

Range = Set of even natural numbers

Codomain = Set of Natural numbers = N

teachoo.com

Example 11
Examine each of the following relations given below and state in
each case, giving reasons whether it is a function or not?
(i) R = {(2, 1), (3, 1), (4, 2}}

In function, first element is not repeating
The first elements are 2,3 and 4
All these are not repeating.
Hence, they have unique images.
So, this relation is a function.

Example 11
Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?
(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

Example 11
Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?
(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

Example 12 teackoo.com
Let N be the set of natural numbers. Define a real valued function
f:N—>N by f (x) = 2x + 1. Using this definition, complete the table
given below.
x | 1 2 3 4 5 6 7
F1)= — f(2)= —(3)= —HAY= AOS) =—(6)= A(T) =
Given f(x) = x?
To complete the table, we put values of x in f(x)
f(1) = 2(1)+1 | £(2)=2(2)4+1 (3) = 2(3) +1 (4) = 2(4) +1
=2¢1 =44+1 =6+1 =8+1
=3 =5 =7 =9

Example 13 teachoo.com

Define the function f: R > R by y = f(x) = x2, x € R. Complete the

table given below by using this definition. What is the domain

and range of this function? Draw the graph of f.

pox | 4-3-2 -1 0 1 2 3 4

y = f(x) =? BE

Given f(x) = x?

To complete the table, we put values of x in f(x)
f(-4) = (4) = 16
f{-3) = (-3)=9
f(-2) = (-2)°=4
f-1)=(-4P=1

Example 14 teachoo.com
Draw the graph of the function f :R > R defined by f {x} = x3, x E R.
f(x) =3, x ER
Also, ai —R
Domain is Range isa
arealnumber Real number
We find various values of f(x} by using different values of x
Value of y = f(x) => | Point to be plotted
ry) 0?=0x0x0 =0 (0,0)
1 =1x1x1 =1 (1,1)
2 23=2x2x2 =8 (2,8)
3 33=3x3x3=27 (3,27)
-1 (-1))=-1x-1x-1 =-1 (-1,-1)
-2 (-2)3 =-2x-2x-2 =-8 (-2, -8)
3 (-3)3 =-3x-3x-3 =-27 (-3, -27)

teachoo.com
Example 15
Define the real valued function f : R — {0} > R defined by f (x) = : x
€ R — {0}. Complete the Table given below using this definition.
What is the domain and range of this function?
x
v= fi) => ae
Given f(x) = -, x € R— {0}
Finding f(x) at different values of x
1
f(-2) = a
=-0.5

Example 16 teachoo.com
Let f(x) = xand g(x) = 2x + 1 be two real functions. Find
(F+ 8) x (F8) 6) (fe) (Z) 6)
f0d = x? & g(x) = 2x41
(f + g) (x) = F(x) + g(x)

= (x?) + (2x + 1)

=x? +2x+1,
«(f+ g) (x) =92 + 2n 41

(f — g) (x) = f(x} - g(x)
= (x?) —(2x + 1)

Example 17 teackoo.com
Let f(x) = «/x and g(x) = x be two functions defined over the set
of nonnegative real numbers. Find (f + g) (x), (f— g) (x}, (fg) &)
f
and (5) (x)
(f + g) (x) = f(x) + g(x)
=yX+x,
(fg) (x) = f(x) — g{x)
= /x-x
(fg) x = f(x) * g(x)
= xxx
1
=x2xx!

Example 18 teackoo.com
Let R be the set of real numbers. Define the real function f: ROR by
f(x) = x + 10 and sketch the graph of this function.
f(x) =x+10
Also, f:R >R
nomat is \ nae isa
areal number Real number
0 0+10 =10 (0, 10)
1 1+10 =11 (1, 11)
2 2+10 =12 (2, 12)
-1 -1+10 =9 (-1,9)
-2 -2+10 =8 (-2, 8)

teachoo.com

Example 19
Let R be a relation from Q to Q defined by R= {{a, b):a,bEQ
and a —b € Z}. Show that
(i) (a, a) ER forallacQ
Given R = {(a, b): a,b € Qanda—b eZ}
Hence we can say that
(a, b) is in relation R if

1. a,b €Qi.e. botha & bare insetQ (Given)

2. a—b€Zi.e. difference of a & b is an integer
We need to prove both these conditions for {a, a)
a,a€Q,i.e.aisinsetQ

teachoo.com
Example 20
Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a linear function from Z into
Z. Find f(x).
Since f is a linear function,
Let f(x) =mx+c.
Putting value of x and y in the function
For (1, 1) For (2, 3)

y=mxtc y=mx+c

L=m(1}+c 3=m(2)+c

l=m+tc 3=2m+c

m+c=1 (1) am+c=3_ ..{2)

Example 21 teackoo.com
x? + 3x45
Find the domain of the function f (x) =—;—-~——
x - 5x44
x7 43x45
f (x) = =———_
(x) x?-5x+4
— x +3x45
x? 4x —x +4
_ x? 4+ 3x45
~ x(x — 4) -1(x — 4)
_ x +3xK45
~ (@-4)(x-1)
In real numbers , the denominator cannot be zero
Hence, {x — 4) (x-—1) #0
-« x#4andx41

teachoo.com
Example 22
The function f is defined by
i x<0
f= 41 ,x=0
x+1,x>0
Draw the graph of f {x}.
For x <0, f(x)=1-x
We find the points to be plotted when x <0
41 1-(-1)=1+1 =2 (1,2)
—2 1-(-2)=1+2 =3 ( - 2,3)
3 1-(-3)=1+3 =4 (-3,4)
-4 1-(-4)=14+4=5 (-4, 5)

Misc 1 teachoo
. . . x, OSxS3
The relation f is defined by _ f(x} = fe 3<x<10
. . . x, OSxS2
The relation gis defined by g (x) = fr 2<x<10
Show that f is a function and g is not a function.
Arelation is a function if
* Every element of first set has an image
* Every element of first set has only one image
Finding value of f(x) and g(x) for different values of x

Misc 2 teackoo.com
_ f.1)- fd)
=y2 Fa a
If fo) = x2, find G41 -1)
Let us first find f(1.1) and f(1)
For f(1.1)
f(1.1} = (1.1)
=11x1.1.
=1.21
For f(1)
f(1) = 1?
=1x1=1

Misc 3 teachoo.com
x? t+2x41
Find the domain of the function f (x) =—~—~———
x — 8x +412
x? + 2x41
f(x)=2 >>
x 8x +12
_— ty?
~ x? 2% —6x +12
- ty
~ x(x —2) — 6(% —2)
_— +1)?
~ (x — 6)(x —2)
In real numbers , the denominator cannot be zero
Hence {x — 6) (x-— 2) #0
=>x#6andx#2

Misc 4 teachoo.com
Find the domain and the range of the real function f defined by.
f(x) = ¥(« — 1)
It is given that the function is a real function.
Hence, both its domain and range should be real numbers
Value of _ ea Whether real
x f(x) = va — 1 number
number greater 1 v1 —1=¥V0 =0 Yes
0 V0 —1=v-1 No
-1 vV-1 —1 =<V-2 No
-2 V¥—2 —1=+V-3 No
Here, f(x) is always positive,
Minimum value of f(x) is 0, Maximum can be any value

Misc 5 teackoo.com
Find the domain and the range of the real function f defined
by f {x} = |x-1].
Here we are given a real function
Hence, both domain and range should be real numbers
number
2 |2-1|=|]1]=1 Yes
1 |1-1] =|0| =0 Yes
0 Jo—1| =|-1] =1 Yes
-1 J-1-1] = |-2| =2 Yes
-2 |-2-1] = |-3] =3 Yes
Here, x can Here, f(x) will always
be any be positive
real number
nr 7erp.,

Misc 6 teackoo.com
Let f= {(x, =) :x€R} bea function from R into R. Determine
14+x
the range of f.
f={(x, =) :xER}
14+x x
Domain Range = =e
1
142? 141 2
We find different values ra oe
(-1)? 1 1
; 7
of — for different ae
values of x iia ill a
Value will always be
between0 &1

Misc 7 teachoo.com
Let f, g: R > R be defined, respectively by f(x) = x + 1, g(x) = 2x-3.
Find f+ g, f—gand f
f(x) =x + 1, g(x) = 2x-3
(f + g) (x) = f(x) + g(x)
= (x +1) +(2x-3)
=3x-2
«.(f + g) (x) = 3x-2
(f—g) (x) = f(x) — g(x)
= (x + 1)- (2x- 3)
=xX+1-2x+3
=-x+4
«. (F-g) (xj =-x +4

Misc 8 teachoo.com
Let f = {(1, 1}, (2, 3), (0, -1), (-1, -3}} be a function from Z to Z
defined by f(x) = ax + b, for some integers a, b. Determine a, b.
Given f(x) = ax+b
ie. y=axt+b
Putting values of x and y in f(x)
For (1, 1) For (2, 3)

y=axtb y=ax+b

1=a(1)+b 3=a(2)+b

l=at+hb 3=2at+b

a+b=1. ...(1) 2a+b=3 ...(2)

teachoo.com

Misc 9 - Introduction
Let R be a relation from N to N defined by
R = {(a, b): a, b € Nand a = b?}. Are the following true?
(i) (a, a) ER, for allae N
Isa =a?
Checking for different values of a

1=V?=1

2#2?

343?
Hence, a= a? is not always true

teackoo.com
Misc 10
Let A = {1, 2, 3, 4}, B= {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9),
(3, 1), (4, 5), (2, 11)}. Are the following true?
(i) fisarelation from AtoB
f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}
First elements = 1, 2, 3, 4
All the first elements are in set A
So, first element is from set A
Second elements = 5, 9, 1, 11
All the second elements are in set B
So, second element is from set B

Mise 14 teachoo.com
Let f be the subset of Z x Z defined by f = {(ab, a+ b): a, b € Z}.
Is f a function from Z to Z: justify your answer.
in function, first element is not repeating

f = {(ab, a+b}: a, b € Z}
Domain Range
Finding elements in f
Also, both a & b are integers
Value ofb | Value of ab Value of a + b

0 0 0x0 =0 0+0 =0

0 1 x1 =0 o+1=1

0 2 0x2 =0 0+2 =2
f will have elements (0,0) , (0,1) , (0,2)

Misc 12 teackoo.com
Let A= {9, 10, 11, 12, 13} and let f: A > N be defined by f(n) = the
highest prime factor of n. Find the range of f.
Given f(n) = Highest prime factor of n
And sincen€A , A={9, 10, 11, 12, 13}
Value of n can be only 9, 10, 11, 12, 13
Doing prime factorization Value of n | Highest prime
factor of n
3/9 2(10 14/11 2/12 13]13 9 3
3]3 5|5 1 2/6 1 10 5
1 1 3/3
1 11 1
12 3
Hence, range of f = {3, 5, 11, 13} 13 13