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Empty relation

A relation R on a set A is said to be an empty relation or a void relation if R = φ.

Universal relation

A relation R from A to B is said to be the universal relation if R = A × B.

Identity relation

A relation R on a set A is said to be the identity relation on A if R = {(a, b): a A, b A and a = b}. Thus identity relation R = {(a, a) : a A}. Identity relation on set A is also denoted by IA.

Reflexive relation

A relation R on a set A is said to be reflexive if aRa a A, i.e., (a, a) R a A.
 
Example

Let A = {1, 2, 3}

 

Let R1 = {(1, 1), (2, 2), (3, 3)}

 

R2 = {(1,1), (2, 2), (3, 3), (1, 2), (2,1), (1,3)}

 

R3 = {(2, 2), (2, 3), (3, 2), (1,1)}

 

Here R1 and R2 are reflexive relations on AR3 is not a reflexive relation on A as (3,3)  A, i.e., 3 R 3.

 

 

Note: The identity relation is always a reflexive relation but a reflexive relation may or may not be the identity relation. In the examples given above R1 is both reflexive and identity relation on A but R2 is a reflexive relation on A but not an identity relation on A.

Symmetric relation

A relation R on a set A is said to be a symmetric relation if aRb bRa (whenever aRb, then bRa), i.e., (a, b) R (b, a) R, a, b A.
 
Example

Let A = {1,2,3}

 

Let R1 = {(1,2), (2,1)};

 

R2 = {(1, 2), (2, 1),(1, 3), (3,1)}

 

Here R1 and R2 are symmetric relations on AR3 = {(2, 3), (3, 1), (1, 3)};

 

R3 is not a symmetric relation on A because (2, 3)  R3 and (3, 2)  R3.

 

 

Antisymmetric relation

A relation R on a set A is said to be antisymmetric if aRb and bRa a = b, i.e., (a, b) R and (b, a) R a = b.
 
Thus R is antisymmetric if ab, then both (a, b) and (b, a) cannot belong to R. One of them may belong to R.
 
Example

Let A = {1, 2, 3}

 

Let R1 = {(1, 2), (1, 3), (1, 1)}

 

R2 = {(1, 2)}; R3 = {(1, 2), (2, 1)}

 

Here R1 and R2 are antisymmetric relations on A but R3 is not antisymmetric relation on A.

 

 

Note: A relation which is not symmetric is not necessarily antisymmetric.
 
Example

Let A = {1, 2, 3}

 

Let R = {(1, 2), (2, 1), (2, 3)}; here R is not symmetric.

 

Also R is not antisymmetric because (1, 2)  R and (2, 1)  R but 1 ≠ 2.

 

Transitive relation

A relation R on a set A is said to be a transitive relation if aRb and bRc aRc, i.e., (a, b) R and (b, c) R (a, c) R.
 
Example

Let A = {1, 2, 3}; R = {(1, 2), (2, 3), (1, 3), (3, 2)}

 

R1 = {(2, 3)}; R2 = {(1, 3), (3, 2), (1, 2)}

 

Then R is not transitive relation on A because (2, 3) ∈ R and (3, 2) ∈ R but (2, 2) ∉ R, i.e., 2. R 2. Again R1 is transitive relation on A because (2, 3) ∈ R1 and (3, x) ∈ R1 ⇒ (2, x) ∈ R1 cannot be considered false because there is no (3, x) ∈ R1 so no need to test whether (2, x) ∈ R1 or not. Finally R2 is a transitive relation.

 

Inverse relation

Let R A x B be a relation from A to B. Then the inverse relation of R denoted by R-1 is a relation from B to A defined by R-1 = {(b, a) : (a, b) R}.
 
Thus (a, b) R (b, a) R-1 a A, b B
 
Clearly domain R-1 = Range R and Range R-1 = Domain R.

Equivalence relation

Let A be a non-empty set, then a relation R on A is said to be an equivalence relation if
  1. R is reflexive, i.e., aR a a A
     
    i.e., (a, a) R, a A
  2. R is symmetric, i.e., if aRb then bRa
     
    i.e., (a, b) R (b, a) R, a, b A
  3. R is transitive, i.e., if aRb and bRc, then aRc
     
    i.e., (a, b) R and (b, c) R (a, c) R
Example
  1. Let A = {1, 2, 3}. Let a relation R be defined on A as R = {(1, 2), (1, 1), (2, 1), (2, 2), (3, 3)}. Then R is reflexive, symmetric and transitive. So R is an equivalence relation on A.
  2. Let A be the set of all members in a family having male and female children and R be the relation “is the brother of” on A. Then R is not an equivalence relation on A. Here R is neither reflexive nor symmetric but it is transitive.




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