Mathematics class 12 mind maps

RELATIONS
(1) Types of Relations
Universal Relation
1. Empty Relation
A relation in which no
2. A relation in which each 3. Identity Relation
4. Ref
A relation in which each (a, a
element of A is related to element of A is related to
element is related to a A
any other element of A, i.e., every element of A, i.e.,
R=ϕ  A×A. R=A×A.
{
itself only. I = ( a, a ) , a ∈ A }



Equivalence Relation
5. Symmetric Relation:
6.
Transitive Relation: 7. A relation R in a set A is sa
(a1, a2)R implies that (a1, a2)R & (a2,a3)R implies
be an equivalence relatio
(a2,a1 )R, for all a1,a2A. that (a1, a3)R, for all
R is reflexive,symmetric &
a1, a2, a3A.
transitive.




Antisymmetric: A relation is Irreflexive
Asymmetric Relation 10. antisymmetric if: 11. 12
9. R is irreflexive iff
( x, y ) ∈ R ⇒ ( y, x ) ∉ R • For all x, y ∈ X[( x, y ) ∈ R & ( y, x ) ∈ R] ⇒ x = y ∀a ∈ A, ( ( a,a ) ∉ R )
• For all x, y ∈ X[( x, y ) ∈ R & x ≠ y] ⇒ ( y, x ) ∉ R




2. EXAMPLE: Relation Reflexive Symmetric Asymmetric A
R1
A = {1, 2, 3, 4}. Identify the properties of relations.
R2
R1 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 2,1) , ( 4,3) , ( 4,1) , ( 3, 2 )} R3
R 2 = A × A, R 3 = φ, R 4 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 4, 4 )} R4

R 5 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 4, 4 ) , (1, 2 ) , ( 2,1) , ( 4,3) , ( 3, 4 )}
R5


If A={1,2},a relation R={(1,2)} on A is a transitive relation.
NOTE using the similar argument a relation R = {(x, y) : x is wife of y} is transitive, where as R = {(x, y)




3. PROPERTIES
2. 3. 4. 5
1. R is asymmetric implies R is not symmetric
that R is irreflexive. By does not imply R is R is not symmetric R is n
R is not reflexive does definition, for all antisymmetric. Counter does not imply R is does

, 5. OPERATION ON RELATION
1. R1 − R 2 = {( a, b )∣( a, b ) ∈ R1 and ( a, b ) ∉ R 2 } 2.R 2 − R1 = {( a, b )∣( a, b ) ∈ R 2 and ( a, b ) ∉ R1}

4. R1 ∩ R 2 = {( a, b )∣( a, b ) ∈ R1 and ( a, b ) ∈ R 2 }


PROPERTIES 6. COM
1) If R1 and R2 are reflexive, and symmetric, then R1  R2
is reflexive, and symmetric. RE
2) If R1 is transitive and R 2 is transitive, then R1 ∪ R 2
need not be transitive. Let R1 ⊆ A × B and
counter example: Let A = {1, 2} such that R1 = {(1, 2 )} and R1 , denoted
R 2 = {( 2,1)}.R1 ∪ R 2 = {(1, 2 ) , ( 2,1)} and (1,1) ∉ R1 ∪ R 2 implies that
R1 R 2 = {( a,c
R1 ∪ R 2 is not transitive.

3) If R1 and R2 are equivalence relations, then R1  R2 is
((
an equivalence relation. NOTE
4) If R1 and R2 are equivalence relations on A, R1 (R2  R3) 
• R1 − R 2 is not an equivalence relation (reflexivity fails).
R1 (R2  R3) =
• R1 − R 2 is not a partial order (since R1 − R 2 is not reflexive).
• R1 ⊕ R 2 = R1 ∪ R 2 − ( R1 ∩ R 2 ) is neither equivalence relation R1 ⊆ A × B, R2 ⊆
nor partial order (reflexivity fails)
5) The union of two equivalence relation on a set is not ( R1oR 2 )−1 = R 2
necessarily an equivalence reation on the set.
6) The inverse of a equivalence relation R is an
equivalence relation.


7. EQUIVALENCE CLASS
Equivalence class of a ∈ A is defined as [a] = {x∣( x,a ) ∈ R} , that is all the ele
the relation R.
Example
E=Even integers, O=odd integers.
(i) All elements of E are related to each other and all elements of O are related to each oth
(ii) No element of E is related to any element of O and vice-versa.

, “In mathematics
the art of proposing a question
must be held of higher value than solving it.”
– Georg Cantor




FUNCTION
1 Classification of function


01. Constant function 03. Polynomial function
f(x) = k, k is a constant. y = f(x) = a0 xn + a1 xn-1+...+ an, n is non negative It is d
integer, ai are real constants. Given a0 ≠ 0, n is
the degree of polynomial function f ( x) =

02. Identity function There are two polynomial functions, f ( x) = 1 + x n & f ( x) = 1 − x n Dom
1 1
The function y = f ( x) = x , ∀x ∈ R satisfying the relation: f ( x) ⋅ f   = f ( x) + f  when
x x
Here domain & Range both R where ‘n’ is a positive integer.




2 Exponential function 3 Logarit


f(x) = ax, a > 0, a ≠ 1. f(x) = loga
0<a<1
0<a<1 a>1
y
y y

1
1
x' 1
x' x x' x x
O O O


y' y'
y'
Domain =R, Range = ( 0, ∞ )
Domain =



Proprieties of Log. Functions

1
1. 4 log a (xy) = log a | x | + log a | y | , where a > 0,a ≠ 1 and xy > 0 2. log a x =
log x a
for a > 0



x x
3. log a   = log a | x | − log a | y | , where a > 0,a ≠ 1 and > 0
y y
4. ( )
log a x n = n log a | x


m y
5. log a n x m = log|a| | x |, where a > 0,a ≠ 1 and x > 0 6. x loga = yloga x wher
n

,  x, x > 0
5 Absolute Value Function y = f ( x) =| x |= 
 − x, x < 0



1. | x |2 = x 2 2. x 2 =| x | 3. | x |= max{− x, x} 4.
x −

a +b a −b a +b a −b
5. max(a, b) = + 6. min(a , b ) = − 7. | x + y |≤| x | + | y | 8
2 2 2 2


9. | x − y |=| x | + | y | if xy < 0 10. |x| ⩾ a (is − ve) x ∈ R 11. a < |x| < b ⇒-b ≤ x ≤ -a




| x |
if x ≠ 0
6 Signum Function y = sgn( x) =  x
0 if x = 0
y=
x<




f(x) = [x] the integral part of x,
7 Greatest Integer Function which is nearest & smaller integer




[ x]< x < [ x] + 1 x − 1 < [ x] < x I ≤ x < I + 1⇒ [ x] = I [x
1. 2. 3.


 0, x ∈ I 2 x, x∈I [ x] < n ⇔
[ x] + [− x] = 
∣
−1, x ∈ I , 2 x + 1, , x ∉ I 5.
[ x] ≤ n ⇔ x < n + 1, n ∈ I
6.


 n + 1   n + 2   n + 4  ... [ x ] + [ y ]< [ x + y ] < [ x ] + [ y ] + 1  1 

 + + + =n [x] + x + +