Maths Chapter 1 Relation & Function Notes
CH-1 RELATION & FUNCTION
❈ Relation: Any subset of AXA is called relation from A to A.
i.e ARA or R ⊆ AXA
2. Let A contain m elements then Total relations = 2m2
Eg A = {a,b,c} then total number of relations = 23x3 = 29 = 512
❈ Types of relations: ->
(i) Reflexive relations: A relation R is called reflexive from A to A
If (a,a) ∈ R ∀ (a,a) ∈ A
Ex A = {1,2,3}
AXA = {(1,1) (1,2) (1,3) (2,2) (2,3) (3,1) (3,2) (3,3)}
Reflexive relation {(1,1) (2,2) (3,3)}
(ii) Symmetric relation: A relation R is symm. If (a,b) ∈ R then
(b,a) ∈ R
Ex R = {(3,2) (2,3) (1,3) (3,1)}
Ex: L₁ || L₂ then L₂ || L₁Let
❈ Transitive relation
A relation is called transitive If (a,b)∈R and (b,c)∈R then (a,c)∈R.
Ex. l₁||l₂ and l₂||l₃ then l₁||l₃
❈ Equivalance relation
A relation ‘R is equivalance if it is i) Reflexive ii) Symmetry
iii) Transitive.
❈ Equivalance class
It is a collection of all element associated with element.
Ex R={(b,b) (a,c) (a,d) (ef)]
equivalance class of a = [b, c, d]
Note: Total no. of reflexive relation in a set contain ‘n’element
= 2ⁿ²⁻ⁿ = 2ⁿ⁽ⁿ⁻¹⁾
2. Total No. of symmetric relations
= 2^(n(n+1)/2)
3 Total No. of transitive relations
= 2ⁿ-1
✤ Function
❈ Function: It is a rule by which two elements are connecting to each other.
It is represented by f(x), f:A→B
Here A is called domain
B is called Co-domain (Range)
❈ Type of function
1. ONE-ONE function (injective): A function is one-one If each element (every) of domain has single image in Co-domain
Steps to check one-one:
1) let x1, x2 ∈ domain
2) find f(x₁) and f(x₂)
3) Put f(x₁) = f(x₂).
4) solve it, If we get x₁ = x₂
Then f(x) is one-one.
If n(A)=N, n(B) = M, Total injective = MPN
2. ON-TO function. (surjective) : A function is onto if every
element of co-domain has pre-image in domain.
Eg :
# steps to check surjective:
1. Put y = f(x) 2. find x intern of y
2. ∀ y ∈ co-domain, x should be in domain.
Ex f(x): 2x-3, f: R→N.
Sol. Let f(x)=y
y=2x-3
x = y+3/2
we have y ∈ N then y+3/2 ∈ R
=> x ∈ R
So it is onto.
3. ONE-ONE Onto (Bijective):→ A function f(x) is Bijective if
it is one-one and onto.
§ If f: A→B, A & B have same number of elements (N). Total
Bijection functions = N!
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