### 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....

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__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 2
^{m.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 = 2^{2.2}= 2^{4}= 16.

**- Basically relations are of 6 types which are as follows-**

__TYPES OF RELATIONS__- Reflexive
- Symmetric
- Anti-Symmetric
- Transitive
- Equivalence
- 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.

- Reflexive- (a,a) ∈ R ∀ a∈A.
- Symmetric - (a,b)∈R ⇒ (b,a)∈R.
- 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.

- Reflexive- (a,a) ∈ R ∀ a∈A.
- Anti-Symmetric - (a,b)∈R and (b,a)∈R, then a=b
- 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 is basically inverse of a relation. Ek relation R ka inverse relation R**

__INVERSE RELATION__^{-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) }.**- 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,**

__COMPLEMENT OF A RELATION__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 R

^{c}= { (1,b),(2,a) }.Best of Luck Students,

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