3. If A and B are two sets then their cartesian product is
denoted by A * B.
It is defined as below
A*B= { (x,y) | x ∈ A and y ∈ B}
If A= { 1,2,3} B = { x,y} then
A* B = {(1,x), (2,x). (3,x), (1,y), (2,y), (3,y) }
B* A={ (x,1), (x,2), ( x,3), (y,1), (y,2), (y,3) }
It is observed that A*B ≠ B*A
4. Relation R from set A to B is subset of A* B
If A= { 1,2,3} B = { x,y} then
R1 = {(1,x), (2,x). (3,x), (1,y), (2,y), (3,y) }
R2 = { (2,x) (2,y)}
are the subset of A* B. Hence R1 & R2 are relation
from A to B.
Relations are denoted by R.
6. Relation R on the set A is said to be reflexive if
a R a, for all a in A.
If If A= { 1,2,3} then
R1 = {(1,1), (2,2), (3,3), (1,2), (2,1) }
is reflexive relation.
Since 1 R1 1
2 R1 2
3 R1 3
7. Relation R on the set A is said to be Symmetric
if a R b implies b R a for a, b in A.
If A= { 1,2,3} then
R1 = {(1,1), (2,2). (3,3), (1,2), (2,1) }
is symmetric relation.
Since 1 R1 2
2 R1 1
1 R1 1
Symmetric Relation.
8. The Relation R on the set A is said to be
transitive if
a R b, b R c implies aR c for a,b,c in A.
If A= { 1,2,3} then
R1 = {(1,1), (2,2), (3,3), (1,2), (2,3),(1,3)}
is transitive relation.
Since 1 R1 2
2 R1 3
1 R1 3
Transitive Relation.
9. The Relation R on the set A is said to be
equivalence if R is reflexive, symmetric and
transitive on A.
If A= { 1,2,3} then
R1 = {(1,1), (2,2), (3,3), (1,2), (2,1)}
Then R1 is equivalence relation on A.
Equivalence Relation.
10. EXAMPLE 1
ℝ: Set of real numbers.
Define the relation R on ℝ as follows:
For a, b in ℝ, a R b ⇔ a ≤ b
Determine whether R is reflexive, symmetric, transitive,
equivalence ?
11. Solution
1. R is reflexive because :
a ≤ a, ∀ a ∈ ℝ
⇒ a 𝑹 a
2. R is not symmetric because:
a ≤ b ⇏ b ≤ a.
Hence, a Rb ⇏ bRa.
3. R is transitive because:
a ≤ b & b ≤ c ⇒ a ≤ c
Hence, a Rb , bRc ⇒ a Rc.
4. Since R is reflexive, transitive but not symmetric, R is
not equivalence relation
Solution
