Day 35: Relations

Hello Dear Students,
Hope you all are doing good.

Aaj hum set relations ke baare mein study karenge,

Let's get started....

RELATIONS
  • Relations basically derives from Cartesian Product. 
  • For example, let A={1,2} and B={a,b}, then, Cartesian product AxB = { (1,a),(1,b),(2,a),(2,b) }. 
  • So, ab hum AxB mein se Relation ko choose karenge, So, relation can be { (1,a),(2,a),(2,b) }. 
  • So in simple words for understanding, 2 sets ke Cartesian product se relation derived hota hai. 
  • Formal definition, Let A and B be 2 non-empty sets, then a relation R from A and B is a subset of AxB. R⊆AxB.
  • Important point to be considered that, Rmax = AxB , that is the maximum number of elements in relation is obviously as same as AxB, the cartesian product, and
  • Rmin = {Φ}, phi means empty set. So, because phi is subset of every set, so the minimum number of elements or sets in relation is Φ.
  • The total number of relations can be 2m.n , where m=number of elements in set 1, and n= number of elements in set 2. In above example, m=2(total elements of A) and n=2(total elements of B), so the total number of relation = 22.2 = 24 = 16. 

TYPES OF RELATIONS - Basically relations are of 6 types which are as follows-
  1. Reflexive
  2. Symmetric
  3. Anti-Symmetric
  4. Transitive
  5. Equivalence
  6. Partial Order 

1. REFLEXIVE RELATION - Let R be a relation in a set A, then R is called reflexive relation if (a,a) ∈ R ∀ a∈A.
For example, Let A={1,2,3,4}, then, 
if R={ (1,1),(2,2),(1,3),(3,3) }, then this relation is not reflexive because isme (1,1),(2,2),(3,3) hai but (4,4) nahi hai. And,
if R={ (1,1),(1,3),(2,2),(3,3),(4,4),(3,4) }, then this relation is reflexive because it has (1,1),(2,2),(3,3),(4,4). 

Note that, if R=Φ, then it is not a reflexive relation, and
if R=AxA, then it is a reflexive relation.

It is also known as Identity Relation.


2. SYMMETRIC RELATION - Let R be a relation in set A, then R is called symmetric relation if (a,b)∈R ⇒ (b,a)∈R or xRy = yRx.
For example, Let A={1,2,3,4}, then,
if R= { (1,2),(2,1),(3,4),(4,3) }, then this is symmetric relation because all (a,b) have its (b,a) in R.
if R= { (1,2),(2,1),(3,4),(2,2) }, then this is not symmetric because (3,4) is there but (4,3) is not there in R.
if R is transitive only then it is also symmetric, for example, if R= { (1,1),(2,2),(3,3),(4,4) }, then R is symmetric relation.

Note that, if R=Φ, then it is a symmetric relation, and
if R=AxA, then it is also symmetric relation.


3. ANTI-SYMMETRIC RELATION - Let R be a relation in set A, then R is called anti-symmetric relation if (a,b)∈R and (b,a)∈R, then a=b. Jaise symmetric relation mein its compulsory that if (a,b)∈R ⇒ (b,a)∈R but in anti-symmetric relation it is not compulsory, and agar (a,b) ka (b,a) hoga relation mein then a=b hoga.
For example, Let A={1,2,3,4}, then,
if R= { (1,2),(3,3),(4,3) }, then it is anti-symmetric because (3,3) is symmetric, and (3,3) ka reciprocal is (3,3) in which 3=3 which is true.
if R= { (1,2),(2,1) }, then it is not anti-symmetric because (1,2) (2,1) is symmetric but 1≠2 and 2≠1. 


4. TRANSITIVE RELATION - A relation R is said to be transitive relation on set A if (a,b)∈R, (b,c)∈R, then (a,c)∈R.
For example, Let A={1,2,3,4}, then,
if R= { (1,2),(2,1),(1,1),(2,2) }, then it is transitive relation because (1,2),(2,1) then (1,1). and also (2,1),(1,2) then (2,2).
if R= { (1,1),(2,2),(3,3) }, then it is transitive relation.
if R= { (1,2) }, then it is transitive relation.
if R = { (3,1),(2,3) }, then it is not transitive because if (2,3),(3,1) then there must be (2,1).


5. EQUIVALENCE RELATION - Equivalence relation is the relation which is a reflexive, symmetric and transitive relation. Jab ek relation given hota hai then wo relation equivalence hota hai jab wo relation reflexive, symmetric and transitive relation ho. 
  1. Reflexive- (a,a) ∈ R ∀ a∈A.
  2. Symmetric - (a,b)∈R ⇒ (b,a)∈R.
  3. Transitive - (a,b)∈R, (b,c)∈R, then (a,c)∈R.
For example, Let A={1,2,3,4}, then, 
R = { (1,1),(1,2),(2,1),(2,2),(3,3),(4,4) }, then it is an equivalence relation because R is reflexive, symmetric and transitive relation. 


6. PARTIAL ORDER RELATION - Partial Order relation basically wo relation hota hai jo reflexive, anti-symmetric and transitive relation ho. 
  1. Reflexive- (a,a) ∈ R ∀ a∈A.
  2. Anti-Symmetric - (a,b)∈R and (b,a)∈R, then a=b
  3. Transitive - (a,b)∈R, (b,c)∈R, then (a,c)∈R.
For example, Let A={1,2,3,4}, then, 
R={ (1,1),(2,2),(3,3),(4,4) }, then it is a partial order relation because R is reflexive, anti-symmetric and transitive relation. 

Note that, partial ordering ko graphically represent kiya ja sakta hai, in which all arrowheads are pointing upwards are there, it is known as Hasse Diagram


INVERSE RELATION - Inverse relation is basically inverse of a relation. Ek relation R ka inverse relation R-1 jisme elements ko inverse kar diya jata hai basically. For example, R={ (1,2),(2,3),(4,4) }, then 
R-1 = { (2,1),(3,2),(4,4) }.

COMPLEMENT OF A RELATION - Agar ek relation R hai so uska complement find karne ke liye hun sabse pehle cartesian product karenge then R ke elements ko consider nahi karenge and uske baaki elements R ke complement honge. For example,
Let, A={1,2} and B={a,b}. and R={ (1,a),(2,b) }
So ab R c ko find karne ke liye sabse pehle find out karenge  Cartesian product AxB = { (1,a),(1,b),(2,a),(2,b) }.
Then R= { (1,a),(2,b) }, so Rc= { (1,b),(2,a) }.


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