What is relations? What are Reflexive, Symmetric and Antisymmetric properties?

Relation is a collection of ordered pairs. For each pair (x, y), each object X is from the symbols of the first set and the Y is from the symbols of the second set.

But a relation can be between one set with it too. In this case the X and Y objects are from symbols of only one set, this case is most common!

For matrixes representation of relations, each line represent the X object and column, Y object. Each square represents a combination based on symbols of the set. The squares are 1 if your pair exist on relation. But it depends of symbols set, maybe it can not use letters, instead numbers or whatever other set of symbols.

• Examples

  Relation R in set A

• Reflexive

  (a, a) ∈ R → a ∈ A

  
  Example:

  set: A = {1,2,3}
  R = {(1,1) (2,2) (3,2) (3,3)}

  

• Symmetric

  (a, b) ∈ R → (b, a) ∈ R

  
  Example:

  set: A = {1,2,3}
  R = {(1,2) (2,1) (2,3) (3,2)}

  

• Anti-Symmetric

  (a, b) ∈ R ^ (b, a) ∈ R → (a = b)

  
  Example:

  set: A = {1,2,3}
  R = {(1,1) (2,2)}

  

• Transitive

  (a, b) ^ (b, c) ∈ R → (a, c) ∈ R

  
  Example:

  set: A = {1,2,3}
  R = {(1,1) (2,2) (1,2) (2,1)}