Frfgtryuygfeefghhjjjhggggggggggggfgggggghhhhgtffffggggggggggggggggggggg

1. Relation
s

2. Definition of Relation
The relations define the connection between the two given sets.
Let A and B be two non empty sets, then a relation R from A to B is a subset
such that R A x B, (a, b) R, aRb.
⊂ ∈
Example:
A={1, 2} B={3, 4}
AxB={(1, 3), (1,4), (2,3), (2,4)}
R=1R3
R=1R5

3. Another Example
Let R be a relation given by R= {(a, b): a=b-2, b>6}. Choose the correct option.
a. (2,4) R
∈ b. (3, 8) R
∈ c. (6, 8) R
∈
Solution:
b. (2,4) R
∈ b. (3, 8) R
∈ c. (6, 8) R
∈
a= 2, b=4 a= 3, b=8 a= 6, b=8
a=b-2 a=b-2 a=b-2
2=4-2 3=8-2 6=8-2
2=2 3=6 6=6
b>6 b>6 b>6
4>6 8>6 8>6

4. Domain and Range
Domain- the set of first values in an ordered pair of a relation.
Range- the set of second values in an ordered pair of a relation.
Example:
If R= {(a, b), (c, d), (e, f)}, then
D={a, c, e}
R={b, d, f}

5. Types of
Relation

6. Trivial
Relation
s 01

7. Trivial Relations
Empty
Relation
Universal
Relation

8. Empty Relation
A relation R in a set A is called empty relation if no element A is related to any element of A.
R=∅ A x A
⊂
Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
a) R= {(a, b): a + b =8} b) R= {(a, b): b=5a} c) R= {(a, b): a & b both even}
Sol. (1, 1): 1+ 1=2 Sol. (1, 2): 2=5(1) 2=5 R={2, 2}
(3, 3): 3 + 3 =6 (2, 3): 3=5(2) 2=10
R= ∅ R= ∅

9. Universal Relation
A relation R in a set A is called universal relation if each element of A is related to every element of A.
R= A x A
Example: Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
a) R= {(a, b): |a-b| 0
≥ } R= {(a, b): a + b 6}
≤ c) R= {(a, b): a & b both odd}
Sol. (3, 3): |3-3| 0
≥ Sol. (3, 3): 3 + 3 6
≤ R={(1, 1), (1, 3), (3, 1), (3, 3)}
(1, 3): |1-3| 0
≥ (2, 3): 2 + 5 6
≤
R= A x A R= A x A

10. Reflexive
Relation
02

11. Reflexive Relation
If (a, a) R for every a A
∈ ∈
Example: Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
a) R= {(a, b): a=b} R={(1, 1), (2, 2), (3, 3)} b) R={(1, 1), (2, 2), (3, 3), (3, 1)}
c) R={(2, 2), (3, 3)} d) R={(1, 3), (1, 1), (2, 1), (2, 2), (3, 3)}

12. Symmetri
c
Relation 03

13. Symmetric Relation
If (a, b) R, then (b, a) R for all a, b A.
∈ ∈ ∈
Example: Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
a) R={(1, 1), (1,3), (3, 1), (2, 3), (2, 2), (3, 2)} b) R={(1, 1), (2, 2), (3, 3)}
c) R={(1, 1), (2, 2), (3, 3), (3, 2)} d) R= A x A
e) R={(1, 2), (3, 2)} f) R={(1, 1)}
g) R={(1, 2), (3, 2), (2, 3)}

14. Transitive
Relation
04

15. If (a, b) R and (b, c) R, then (a, c) R for all a, b, c A
∈ ∈ ∈ ∈
Example: Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
a) R={(1, 2), (2, 3), (1, 3)} b) R={(1, 3), (2, 3)}
c) R= A x A d) R={(1, 2), (2, 3)}
Transitive Relation

16. Another Example
Example: Example: Let R be a relation in a set A
A={1, 2, 3, 4}
A x A={1, 2, 3, 4} x {1, 2, 3, 4}
A x A={(1, 1), (1, 2), (1,3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4,
4)}
a) R={(1, 2), (2, 3), (1, 3), (4, 3)}
b) R={(2, 3), (4, 3), (1, 1)}
c) R= {(1, 2), (2, 3), (1, 3), (3, 4), (2, 4)}
a=1 b=2, b=2 c=3, a=1 c=3 TR
A=2 b=3, b=3 c=4, a=2 c=4 TR
A=1 b=3, b=3 c=4, a=1 c=4 NTR

17. 05
Equivalence
Relation

18. A relation is an Equivalence Relation if it is reflexive, symmetric, and
transitive.
Example: Example: Let R be a relation in a set A
A={1, 2, 3}
A x A={1, 2, 3} x {1, 2, 3}
A x A={(1, 1), (1, 2), (1,3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
1. R= A x A
Equivalence Relation