Relation And Function of class 12th .pdf

It is kind of mind map

1. Relations
and Functions
: is both one-one
fx
y
→
and onto, then is bijective.
f
f
3
is bijective.
A binary operation '*' on a set A is a function *
:×
A denoted by *
AA
ab
i.e.
→∀
A, * A. Commutative if
*=
a,b,
ab ab
b
∈∈
*.
Associative if
(*
)
*=
a a,b
Aa
bc
a
∀∈
*
(*
),
A. is identi y if =
bc
a,b
eA
a *
ea
∀∈
∈
t
=*
and is the inverse
ea
aA
.b
A
∀∈
∈
of , if
.*
==
*.
Addition is a
aA
ab
eb
a
∈
binary operation on the set of integers.
A function
:i
s invertible, if a function
fx
y
→∃
gy
x gof
Io
gI
: such that = and
=.
Then,
→
xy
f
gf
f
is the inverse of . If is invertible, then it is both
one-one and onto and vice-versa.
Fo
r.
If
()
=
eg
fx
xf
NN
f
an
d:
,t
he
ni
s
invertible
.
→
If
:,
:z
and
:z
s
Theorem
1:
fx
yg
yh
→→
→
ar
ef
unctions
,t
he
n(
)=
()
.
ho go
th
og
fo
Let : and
:b
e two
Theorem
2:
fx
yg
yz
→→
invertible function , then is invertible ands
gof
()
=o
.
gof
fg
–1 –1 –1
A relation
:i
s
RA
A
→
reflexive if
aR
aa
A
∀∈
Equivalence relation
(reflexive, symmetric, transitive ,
e.g.
Let = the set of all triangles in a
T
plane and : defined by
RT
T
→
= {( , )} : is congruent to }.
RT
TT
T
12
12
Then, is equivalence.
R
The composition of functions :
fA
B
→
and
:i
s denoted by , and is
gB
C gof
→
defined as : given by =
gof
AC
gof(x)
→
gf
xx
A e.g.
AN
fg
((
)) . let = and
,:
∀∈
such that
()
= and
()
=
NN
fx
xg
xx
→
23
∀∈
xN
gof
gf
g
. Then (2)
=(
(2))
=(
2)
2
=
4=
64.
3
A relation : is universal
RA
A
→
if
,=
×.
a R b a,b A,R
AA
∀∈
if
=,
then R is universal.
R
φ
A relation : is symmetric
RA
A
→
if ,
aRb bR
aa
bA
⇒∀
∈
A relation × is transitive
R :
AA
if
,.
aRb bRc aRc a,b,c A
⇒∀
∈
}
Trivial
Relations
+ +
:i
s onto it for energy
fx
y
→
yx
X f(x)
yf
, S.t.
=,
is onto.
∈∃
∈
Y
2
fx
y
: is one-one if
→
()
=(
)=
fx fx
xx
12
12
⇒
∀∈
xx
x
12,.
Other wise,
ff
is many- one, is one-one.
1
f
2
f
1
f
3
123
a bc
123
a
b
1 23
a cb
d
A relation
:A
is empty
RA
→
if
.=
.
ab
a,
bA
RA
AR
∀∈
φ
⊂
×
Fo
r:
= {( )
:=
}, = {1,5,10}
eg R a,b
ab
A
2
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