Page 15 - Discrete Structure II

Exercise II

Which of the following relations in the A = { a, b, c, d} is reflexive

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

R2= { (a, a), (a, b), (b, b) , (c, c), ( d, d) ( c, d) }
R3= { (a, a), (a, b), (b, b) , (c, c), ( d, d) ( c, d), (d, c) }

R3= { (a, a),(b, b) , (c, c), ( d, d) }

Answer

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

Not reflexive, because (d, d) ∉ R1

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

Yes, it is reflexive

R3= { (a, a), (a, b), (b, b) , (c, c), ( d, d) ( c, d), (d, c) }

Yes, it is reflexive
R3= { (a, a),(b, b) , (c, c), ( d, d) }

Yes, it is reflexive

Symmetry

Let R be a relation on a set A, R is said to be symmetry if a R b whenever b R a, that is (a, b) ( , ) ∈
( , ) ∈
∀ , ∈ , ( , ) ∈ ( , ) ∈

Example I

Let A = {2, 5, 8, 9 }. Which of the following relation on A is symmetric

1. R1 = { (2, 2) , (3, 4), ( 8 , 9 ) , (5, 2) , (5, 5)}
2. R2 = { (2, 2) , (3, 4), ( 8 , 9 ) , (4, 3) , (5, 5), (8, 8) , (9, 8)}
3. R3 = { (2, 5) , (3, 4), ( 8 , 9 ) , (4, 3) , (5, 5), (5, 2) , (8, 8) , (9, 8)}
4. R4 = { (2, 5) , (5, 2) , (8, 8) , (9, 8)}