Page 22 - Algebra 1

Different types of relations
❖ Reflexive relation
Every element maps to itself.
If set P = {2,4}
A reflexive relation R = {(2, 2), (2, 4), (4, 4), (4,2)} In general, it’s given by (p, p) ∈ R
❖ Symmetric relation
In a symmetric relation if p = q, the q = p
For example, if p||q, then q||p
In general, it’s given by (p, q) ∈ R ⇒ (q, p) ∈ R
❖ Transitive relation
In a transitive relation if p = q, and q = r, then p = r
In general, it’s given by (p, q) ∈ R, (q, r) ∈ R ⇒ (p, r) ∈ R
❖ Equivalence relation
When a relation is reflexive, symmetric, and transitive, it’s called an equivalence relation.
Worked Example 9
P = {1, 2, 3, 4, 5} is defined as R = {(𝑝, 𝑞): 𝑞 = 𝑝 + 2}. Is the relation reflexive, symmetric, transitive or equivalence?
Solution:
R = {(𝑝, 𝑞): 𝑞 = 𝑝 + 2}
R={(𝑝, 𝑝+2) ∈ {1,2,3,4,5}} ={(1, 3),(2,4),(3,5),(4,6),(5,7)}
R is not symmetric because (p, q) ∈ R but (q, p) ɇ R
R is not transitive because (p, q) ∈ R, (q, r) ∈ R but (p, r) ɇ R For example, (1, 3) ∈ R, (2, 4) ∈ R but (3, 5) ɇ R
R is not reflexive since (p, p) ɇ R
The relation R of Set
Page 21 of 54
ALGEBRA