Summary - Mathematics

RELATIONS
(1) Types of Relations
Universal Relation
1. Empty Relation
A relation in which no
2. A relation in which each 3. Identity Relation
4. Ref
A relation in which each (a, a
element of A is related to element of A is related to
element is related to a A
any other element of A, i.e., every element of A, i.e.,
R=ϕ  A×A. R=A×A.
{
itself only. I = ( a, a ) , a ∈ A }



Equivalence Relation
5. Symmetric Relation:
6.
Transitive Relation: 7. A relation R in a set A is sa
(a1, a2)R implies that (a1, a2)R & (a2,a3)R implies
be an equivalence relatio
(a2,a1 )R, for all a1,a2A. that (a1, a3)R, for all
R is reflexive,symmetric &
a1, a2, a3A.
transitive.




Antisymmetric: A relation is Irreflexive
Asymmetric Relation 10. antisymmetric if: 11. 12
9. R is irreflexive iff
( x, y ) ∈ R ⇒ ( y, x ) ∉ R • For all x, y ∈ X[( x, y ) ∈ R & ( y, x ) ∈ R] ⇒ x = y ∀a ∈ A, ( ( a,a ) ∉ R )
• For all x, y ∈ X[( x, y ) ∈ R & x ≠ y] ⇒ ( y, x ) ∉ R




2. EXAMPLE: Relation Reflexive Symmetric Asymmetric A
R1
A = {1, 2, 3, 4}. Identify the properties of relations.
R2
R1 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 2,1) , ( 4,3) , ( 4,1) , ( 3, 2 )} R3
R 2 = A × A, R 3 = φ, R 4 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 4, 4 )} R4

R 5 = {(1,1) , ( 2, 2 ) , ( 3,3) , ( 4, 4 ) , (1, 2 ) , ( 2,1) , ( 4,3) , ( 3, 4 )}
R5


If A={1,2},a relation R={(1,2)} on A is a transitive relation.
NOTE using the similar argument a relation R = {(x, y) : x is wife of y} is transitive, where as R = {(x, y)




3. PROPERTIES
2. 3. 4. 5
1. R is asymmetric implies R is not symmetric
that R is irreflexive. By does not imply R is R is not symmetric R is n
R is not reflexive does definition, for all antisymmetric. Counter does not imply R is does