RElAtIoNs & FUnCtIoNs
LEcTuRe 1
A 1
b 2
C 3
d 4
A B
z Ith ke basics
muumuu
Two sets :
Input set , output set
t
Cartesian product of 2 sets is AXB →
output set
t
input
set
here element A maps to
each
of
element B
every of .
For Ln : I a
2 b A =L 1.2.34 ) s { 13123133443
3 c B -
-
Sa b. e3 ,
4,21 41,33
Y
B
A
A- XB is { (I ,
a ) ,
d. b 1,11 ,
c ) ,
( 2. a) 12 ,
b) ,
12 ,
c) ,(3. at ,( 3 b)
, ,
( 3. c) .
( Y at
, ,
4. b) 14,41
Now any subset of AXB is a Relation from A to B .
for En s
s cha ) ,
(2. a ) ,
( 3. all is a RELATION .
, •
In a relation ,
one input can have multiple outputs .
•
In a relation R in the form ( n , y) ,
n is element from input set &
y is from output set .
•
Also IM me
padha tha ki a relation is called a Function if input
has unique output .
'm
12 ke relation me hum A XA me Relations dekhen.ge .
And But
types of relations padhenge . use
peke knock basic questions bar eete hai -
Q, Set A is { 1,2 , 3 , 4 } & R & P are relations on AXA such that ,
R = ( ( 2,1 ) (2 , ,
1) ,
(1,2 ) ,
12,31 14 , 1) ,
/
P =
I ( 2,41 , 6,4 ) , 14 ,
I 1 , 4,213
which of these relations is a
function ? ?
Are In R ,
I is related to 1 & 2 ,
ie input 14 has 2 outputs 142 ie not
a function .
In P .
Each of the input 1,2 , 3,4 has only L output ,
ie P is a function .
*
Important :
If a relation defined on A XB , if the relations has more elements
than no .
of elements of A then ,
it CANNOT be a
function .
Par Kyun ? ?
Dekho A
input set hai .
Agar A me in elements hai to manimum
case me bhi agar Koi relation ho jisme Saare elements of A are
mapped &
function is created then only in pairs will enlist .
*
Q, How many number of relations can be defined from set A of m elements ?
d
AXA
A- No AXA
.
of relations = no .
of subsets of
No .
of elements in AXA = mxm = m2
Now ,
no .
of subsets = 2K ( where k = no .
of elements in that set )
So ,
It no .
of relations are possible .
Proof [No of .
subsets in a set ] : P = { a , .az . - .
an }
{ a } 1923 . .
.
.
. San } La .az } Ear 933 ,
.
. . .
n
"
Cn
"
C, subsets Cz
→ Moth C , th Cz th Cz .
. .
- { a , ,
Az 0373 .
. -
E )
,
→ 20h ' I '
h
Cz
LEcTuRe 1
A 1
b 2
C 3
d 4
A B
z Ith ke basics
muumuu
Two sets :
Input set , output set
t
Cartesian product of 2 sets is AXB →
output set
t
input
set
here element A maps to
each
of
element B
every of .
For Ln : I a
2 b A =L 1.2.34 ) s { 13123133443
3 c B -
-
Sa b. e3 ,
4,21 41,33
Y
B
A
A- XB is { (I ,
a ) ,
d. b 1,11 ,
c ) ,
( 2. a) 12 ,
b) ,
12 ,
c) ,(3. at ,( 3 b)
, ,
( 3. c) .
( Y at
, ,
4. b) 14,41
Now any subset of AXB is a Relation from A to B .
for En s
s cha ) ,
(2. a ) ,
( 3. all is a RELATION .
, •
In a relation ,
one input can have multiple outputs .
•
In a relation R in the form ( n , y) ,
n is element from input set &
y is from output set .
•
Also IM me
padha tha ki a relation is called a Function if input
has unique output .
'm
12 ke relation me hum A XA me Relations dekhen.ge .
And But
types of relations padhenge . use
peke knock basic questions bar eete hai -
Q, Set A is { 1,2 , 3 , 4 } & R & P are relations on AXA such that ,
R = ( ( 2,1 ) (2 , ,
1) ,
(1,2 ) ,
12,31 14 , 1) ,
/
P =
I ( 2,41 , 6,4 ) , 14 ,
I 1 , 4,213
which of these relations is a
function ? ?
Are In R ,
I is related to 1 & 2 ,
ie input 14 has 2 outputs 142 ie not
a function .
In P .
Each of the input 1,2 , 3,4 has only L output ,
ie P is a function .
*
Important :
If a relation defined on A XB , if the relations has more elements
than no .
of elements of A then ,
it CANNOT be a
function .
Par Kyun ? ?
Dekho A
input set hai .
Agar A me in elements hai to manimum
case me bhi agar Koi relation ho jisme Saare elements of A are
mapped &
function is created then only in pairs will enlist .
*
Q, How many number of relations can be defined from set A of m elements ?
d
AXA
A- No AXA
.
of relations = no .
of subsets of
No .
of elements in AXA = mxm = m2
Now ,
no .
of subsets = 2K ( where k = no .
of elements in that set )
So ,
It no .
of relations are possible .
Proof [No of .
subsets in a set ] : P = { a , .az . - .
an }
{ a } 1923 . .
.
.
. San } La .az } Ear 933 ,
.
. . .
n
"
Cn
"
C, subsets Cz
→ Moth C , th Cz th Cz .
. .
- { a , ,
Az 0373 .
. -
E )
,
→ 20h ' I '
h
Cz
