Page 14 - Discrete Structure II

(1, -1) ∈ R2, R3, R6

Properties of relations

a. Reflexivity
b. Symmetry
c. Antisymmetry

d. Transitivity

a. Reflexivity

Definition: A relation R on a set A, is said to be reflexive, if every element in A is in

relation with itself.

Example

Let A = {a, b, c} be a set. Which of the following relations on A is reflexive?

R1 = {(a, b), (a, a), (b, b) (b, c), (c, c)}
R2 = {(b, a), (a, a) ,(b, b) (b, c) , (a, c), (c, a)}

R3= { (a, a) }

R4 = { (a, a) (b, b) ,(b, c) , (c, c)}

R5 = { (a, a) (b, b) , (c, c)}

Answer:

R1 = {(a, b), (a, a), (b, b) ,(b, c), (c, c)} Yes, reflexive because every element is in relation with itself.

R2 = {(b, a), (a, a) ,(b, b) (b, c) , (a, c), (c, a)} no, because c is not in relation with itself
R3= { (a, a) } no, because (b, b) for instance is not an element of R 3

R4 = { (a, a) (b, b) ,(b, c) , (c, c)} Yes, it is reflexive

R5 = { (a, a) (b, b) , (c, c) } Yes, it is reflexive