Name : Jayalakshmi Jagadeesan.
Discussion about Equivalence Relation,Binary Relation, and Partition :
3 properties of Equivalence Relations are
1. congruence A = A
2. If A = B then B =A
3. If A = B and B = C then A = C
->
->
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Reflexive Property
Symmetric Property
Transitive Property
A Partition Theorem
Let X be a non-empty set and ~ an equivalence relation on X. The equivalence
classes of ~ form a partition (a disjoint collection of non-empty subsets whose
union is the whole set) of X.
The reflexive property of equivalence relation says that every element of X is
equivalent to at least one element of X, namely itself. Hence, that element and,
thus, every element, belong to some equivalence class. (Note that this fact is
assumed in the above definition of equivalence class.) Since every element of X
belongs to an equivalence class, X is contained in the union of the equivalence
classes. To finish the proof, one must show that the equivalence classes are
disjoint: for any two different equivalence classes A and B, A is disjoint from B.
Let x be in both A and B. Let a be any element in A and b be any element in B.
Since a and x are in A, a ~ x. Since b and x are in B, x ~ b. Hence by the
transitive property, a ~ b. Thus any element of A is equivalent to any element of
B. Consequently, A and B are the same subset. The upshot of this equality is that
if two equivalence classes have one common element, they are the same
equivalence class. Thus any pair of different equivalence classes are disjoint.
Therefore, the equivalence classes of X form a partition of X.
Also note that if X is empty, ~ and the partition are both empty.
The Correspondence of Equivalence Relation and Partition
Let X be a set. By the partition theorem, every collection of equivalence classes
equivalence is a partition of X. By the converse partition theorem, every partition
is a collection of equivalence classes induced by the mentioned equivalence
relation. One of the most common uses of an equivalence relation is to produce
a partition. Producing an equivalence relation from a partition is less common,
but also useful.
Definition (binary relation):
A binary relation from a set A to a set B is a set of ordered pairs <a, b> where a
is an element of A and b is an element of B.
When an ordered pair <a, b> is in a relation R, we write a R b, or <a, b>
means that element a is related to element b in relation R.
When A = B, we call a relation from A to B a (binary) relation on A.
R. It
Examples:
If A = {1, 2, 3} and B = {4, 5}, then {<1, 4>, <2, 5>, <3, 5>}, for example, is
a binary relation from A to B.
However, {<1, 1>, <1, 4>, <3, 5>} is not a binary relation from A to B because
1 is not in B.
The parent-child relation is a binary relation on the set of people. <Jaya, Jaya
Jr.>, for example, is an element of the parent-child relation if Jaya is the mother
of Jaya Jr.
Generalization Equivalence Realations:
One special case is binary relation
AεUxU
A = { (1,2), (3,4),(2,2),(1,1)}
A = {(1,2), (3,4), (2,2), (1,1)
(2,1),(4,3),(4,4),(3,3)}
There exists 3 representations
1. A = {(1,2), (3,4), (2,2) ,(1,1)
(2,1), (4,4), (3,3)}
2. N(1) = {2}
N(2) = {1}
N(3) = {3,4}
N(4) = {4}
This is called Granular Computing View.
3. 1
-> 2
<3
->
<-
4
This is called Graphical Network Diagram.This network is not joint.