1. Relations

2. Partition of a set Let A be a non-empty set.
Suppose there exist non empty subsets…
A1,A2,A3…….AK of A such that following two conditions hold
A is the union of A1,A2,A3…….AK
Any of two subsets A1,A2,A3…….AK are disjoint that is Ai ∩ Aj = Ø for I not equal to j.
Then the set P= {A1,A2,A3…….AK } is called the Partition of A and A1,A2,A3…….AK are called blocks or cells of the partition.

3. Let A={1,2,3,4,5,6,7,8} subsets A1={1,3,5,7}, A2={2,4}, A3={6,8}
We observe that A is the union of given 3 sets.
Any two of given subsets are disjoint.
So P={A1,A2,A3} is a partition of A with A1,A2,A3 as blocks.

4. Find the equivalence classes of elements of A w.r.t R.
Let A={1,2,3,4} for the equivalence relation R={(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3)(4,4)} defined on A. Determine the partition induced.
Find the equivalence classes of elements of A w.r.t R.
Consider the distinct equivalence classes.
These distinct or disjoint equivalence classes construct the partition.

5. Find the equivalence classes of elements of A={1,2,3,4} w.r.t R.
[1]={1,2}
[2]={1,2}
[3]={3,4}
[4]={2,4}
[1] and [3] are distinct.
P={ [1], [3]} = { {1,2},{3,4} }
A=[1] U [3] and [1] and [3] are dosjoint

6. A={1,2,3,4,5} R={(1,1),(2,2),(2,3),(3,2),(3,3),(4,4),(4,5),(5,4)(5,5)} defined on A. Find the partition of A induced by R.

7. Let A={a,b,c,d,e} P={ {a,b}, {c,d}, {e} } is the partition of A
Let A={a,b,c,d,e} P={ {a,b}, {c,d}, {e} } is the partition of A. Find the Equivalence relation inducing this partition.
Since a,b belongs to same block, we have
aRa
aRb
bRa
bRb
since c,d belongs to same block, we have
cRc, cRd, dRc, dRd
since e belongs to block {e} which contains on e,so we have eRe.
Thus, the required equivalence relation R is given by
R={ (a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d),(e,e) }.

8. On set of all integers Z, a relation R is defined by aRb iff a2=b2
On set of all integers Z, a relation R is defined by aRb iff a2=b2. verify R is an equivalence relation. Determine the partition induced by this relation.
Clearly R is an equivalence relation
For any any a ∈ Z we have
[a] ={ x ∈ Z|(x,a) ∈ R}
={x ∈ Z| x2 = a2}
={x ∈ Z| x= ±a}
[0]={0}, [1]= {1,-1}, [2]= {2,-2}…….[n]={n,-n} for n ∈ Z+ so P={[0],[n]} where n ∈ Z+

9. Transitive Closure
Let R be a relation on a set A. The connectivity relation R* consists of pairs (a, b) such that there is a path of length at least one from a to b in R.
i.e.,
Theorem :The transitive closure of a relation R equals the connectivity relation R*.
Let R be a relation on a set A with |A|=n then
Ch8-9 9

10. Example. Let R be a relation on a set A, where
What is the transitive closure Rt of R ?
Sol :
∴Rt = R  R2  R3  R4  R5
= {(1,2),(2,3),(3,4),(4,5),
(1,3), (2,4), (3,5),
(1,4), (2,5),
(1,5)} 1 3 5 2 4

11. Partial orders
The Relation R on set A is said to be a partial ordering relation or partial order on A if
R is reflexive
R is antisymmetric
R is transitive
Set A with a partial order R defined on it is called a Partial ordered set or an ordered set or Poset and it is denoted by a pair (A,R)

12. The most familiar partial order is the relation “less than or equal to” (Z,  ≤)
The subset relation defined on the power set of a set S is partial ordered on S
(P(s), ⊆)

13. Total Order
Let R be a partial order on set A.
Then R is called a total order on A(Linear order) if for all x,y ∈ A either xRy or yRx.
In this case poset (A,R) is called totally ordered set or chain.
≤ : for any x,y ∈ A we have x ≤y or y ≤x
A={1,2,4,8} Divisibility Relation is totally ordered.

14. Show that the set of all positive integers is not totally ordered by the relation of divisibility.
For a set A to be totally ordered by a partial order relation R, we should have aRb or bRa for every a,b ∈ A.
If R is the divisibilty relation on Z+
aRb or bRa need not hold for every a,b ∈ Z+
if we take a=2,b=3
a does not divide b and b does not divide a
So Z+ is not totally ordered by the relation of divisibility.

15. Hasse Diagram
Since Partial order is a relation on set A, we can think of graph of a partial order if the set is finite.
Drawing of its transitive reduction
Named after Helmut Hasse.

16. Rules for drawing Hasse Diagram
For reflexive relation, At every vertex in the diagraph of a partial order there would be a loop
While drawing the graph of partial order, we need not exhibit such loops explicitly. It will be automatically understood by convention.
For transitive relation we draw edges form a to b, and b to c, then a to c.
No need to exhibit the edge a to c. It will be automatically understood by convention

17. For simplification of a diagraph vertices are denoted by dots.
We draw the edges in such a way that all the edges point upward . No need to put arrows in the edges.
The diagraph of partial order drawn by adopting the conventions indicated above is called Poset diagram/Hasse diagram for partial order.

18. Example
Draw the Hasse diagram representing the partial ordering {(a, b) | a divides b} on {1, 2, 3, 4, 6, 8, 12}.
Sol : 8 12 4 6 2 3 1

19. Example 13.
Draw the Hasse diagram for the partial ordering {(A, B) | A  B} on the power set P(S) where S={a, b, c}.
Sol :
{a, b, c}
{a, b}
{a, c}
{b, c}
{a}
{b}
{c} 

20. 1.Draw the Hasse Diagram for Representing the positive divisors of 36.

21. 2. Let A={1,2,3,4,6,12}. On A define a relation R by aRb iff a divides b. Prove that R is a partial order on A. Draw the Hasse diagram.

22. In the following cases, consider the partial order of divisibility on set A. Draw the Hasse diagram for the poset and determine whether the poset is totally ordered or not.
1) A={ 1,2,3,5,6,10,15,30}
2) A={2,4,8,16,32}

24. The Hasse Diagram of a partial order R on the set A={1,2,3,4,5,6} is given below.Construct the matrix and diagraph of R.

25. Extremal Elements in Posets
Consider a poset (A,R). We define some special elemnts called Extremal elements that may exist in A.
Maximal Element
Minimal Element
Greatest Element
Least Element
Upper bound
Lower bound
Least upper bound(Supremum)
Greatest lower bound(Infimum).

26. We define
Maximal Element
Minimal Element
Greatest Element
Least Element
With reference to the set A as a whole. We define
Upper bound
Lower bound
Least upper bound(Supremum)
Greatest lower bound(Infimum).
With reference to the specified subset of A.

27. An element a of set A is the maximal element of set A if in the Hasse diagram no edge starts at a.
An element a of set A is the minmal element of set A if in the Hasse diagram no edge terminates at a.
An element a of set A is called the greatest element of A if xRa for all x belongs to A
An element a of set A is called the least element of A if aRx for all x belongs to A

28. Find all the minimal and maximal elements for the Posets shown in below Hasse diagrams

29. Find the greatest and least elements

30. An element a belongs to A is called the upper bound of a subset B of A if xRa for all x belongs to B.
An element a belongs to A is called the Lower bound of a subset B of A if aRx for all x belongs to B.

31. Consider the set A={1,2,3,4,5,6,7,8} and the partial order on A as shown below. Consider the subsets B1={1,2} and B2={3,4,5} of A as shown below

32. An element a belongs to A is called greatest lower bound(GLB) of a subset B of A if the following two conditions hold.
a is a lower bound of B
If al is a lower bound of B then al R a.

33. An element a belongs to A is called Least upper bound(LUB) of a subset B of A if the following two conditions hold.
a is a an upper bound of B
If al is an upper bound of B then aRal