Question

3. Given that A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}, how many functions f: A → B are there (i) in total? (ii) which are one-to-one? (iii) with f(2) = 6 and f(3) = 5? (iv) with f(1) = 1 and f(2) # 2?

Answer 1

$\textbf{3.}$ Given set  
A= {1,2,3)
and  
B = {1, 2, 3, 4, 5, 6}
.

So contains three and   contains six elements .

1.
Let  
A B
be any function . Then  
f(1)
has six choices  
1. 23.4.5.6
. Similarly
f(2), (3)
also have six choices  
1. 23.4.5.6
.

Hence total number of function   is ,

A B

= 6 x 6 x 6

= 216

Answer :  
216
.

ii.
If  
A B
be a one-to-one function then   
f(1)
has six choices  
1. 23.4.5.6
.
f(2)
  has five choices as the element we already chosen as  
f(1)
we cannot choose as
f(2)
as the function is one-to-one . Similarly  
f(3)
has four choices .

Hence total number of one-to-one function   is ,

A B

TX9X9=

= 120

Answer :  
120
.

iii.
If  
A B
be a function such that  
f(2) = 6
and  
f(3) = 5
.

Then  
f(1)
have six choices  
1. 23.4.5.6
   and each  
f(2)
,  
f(3)
have 1 choices as  
f(2) = 6
and  
f(3) = 5
.

Hence total number of function  
A B
such that  
f(2) = 6
and  
f(3) = 5
is ,

= 5x1x1

=5

Answer :  $\bg_green 5$  .

iᏙ
  If  
A B
be a function such that  
f(1) #1
and  
f(2) #2
then ,

  
f(1)
has five  choices  
2.3.4.5.6
,  
f(2)
   has five  choices and  
f(3)
has   six choices  
1. 23.4.5.6
.

Hence total number of function   
A B
be such that  
f(1) #1
and  
f(2) #2
is ,

$=5\times 5\times 6$

$=150$

Answer :  

.

.

.

.

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