Relations Partition of a set Let A be

Relations

Partition of a set Let A be a non-empty set. n Suppose there exist non empty subsets… A 1, A 2, A 3……. AK of A such that following two conditions hold ü A is the union of A 1, A 2, A 3……. AK ü Any of two subsets A 1, A 2, A 3……. AK are disjoint that is Ai ∩ Aj = Ø for I not equal to j. Then the set P= {A 1, A 2, A 3……. AK } is called the Partition of A and A 1, A 2, A 3……. AK are called blocks or cells of the partition. n Ch 8 -2

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

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. n Consider the distinct equivalence classes. n These distinct or disjoint equivalence classes construct the partition. n Ch 8 -4

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 n Ch 8 -5

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. Ch 8 -6

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 a. Ra a. Rb b. Ra b. Rb § since c, d belongs to same block, we have c. Rc, c. Rd, d. Rc, d. Rd § since e belongs to block {e} which contains on e, so we have e. Re. § 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) }. n Ch 8 -7

On set of all integers Z, a relation R is defined by a. Rb iff a 2=b 2. verify R is an equivalence relation. Determine the partition induced by this relation. Clearly R is an equivalence relation For any a ∈ Z we have [a] ={ x ∈ Z|(x, a) ∈ R} ={x ∈ Z| x 2 = a 2} ={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+ n Ch 8 -8

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 Ch 8 -9

Example. Let R be a relation on a set A, where A={1, 2, 3, 4, 5}, R={(1, 2), (2, 3), (3, 4), (4, 5)}. What is the transitive closure Rt of R ? Sol : 1 ∴Rt = R R 2 R 3 R 4 R 5 = {(1, 2), (2, 3), (3, 4), (4, 5), (1, 3), (2, 4), (3, 5), (1, 4), (2, 5), (1, 5)} 3 5 2 4 Ch 8 -10

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) n Ch 8 -11

The most familiar partial order is the relation “less than or equal to” (Z, ≤) n The subset relation defined on the power set of a set S is partial ordered on S (P(s), ⊆) n Ch 8 -12

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 x. Ry or y. Rx. 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. n Ch 8 -13

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 a. Rb or b. Ra for every a, b ∈ A. If R is the divisibilty relation on Z+ a. Rb or b. Ra 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. n Ch 8 -14

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. n Ch 8 -15

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 n Ch 8 -16

For simplification of a diagraph vertices are denoted by dots. n We draw the edges in such a way that all the edges point upward . No need to put arrows in the edges. n The diagraph of partial order drawn by adopting the conventions indicated above is called Poset diagram/Hasse diagram for partial order. n Ch 8 -17

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 Ch 8 -18

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} {a, b} {c} {b, c} {b} Ch 8 -19

1. Draw the Hasse Diagram for Representing the positive divisors of 36. . Ch 8 -20

2. Let A={1, 2, 3, 4, 6, 12}. On A define a relation R by a. Rb iff a divides b. Prove that R is a partial order on A. Draw the Hasse diagram. Ch 8 -21

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} n 2) A={2, 4, 8, 16, 32} n Ch 8 -22

Ch 8 -23

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. Ch 8 -24

Extremal Elements in Posets Consider a poset (A, R). We define some special elemnts called Extremal elements that may exist in A. q Maximal Element q Minimal Element q Greatest Element q Least Element q Upper bound q Lower bound q Least upper bound(Supremum) q Greatest lower bound(Infimum). Ch 8 -25

We define q Maximal Element q Minimal Element q Greatest Element q Least Element With reference to the set A as a whole. We define q Upper bound q Lower bound q Least upper bound(Supremum) q Greatest lower bound(Infimum). With reference to the specified subset of A. Ch 8 -26

n n 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 x. Ra for all x belongs to A An element a of set A is called the least element of A if a. Rx for all x belongs to A Ch 8 -27

Find all the minimal and maximal elements for the Posets shown in below Hasse diagrams Ch 8 -28

Find the greatest and least elements Ch 8 -29

An element a belongs to A is called the upper bound of a subset B of A if x. Ra for all x belongs to B. n An element a belongs to A is called the Lower bound of a subset B of A if a. Rx for all x belongs to B. n Ch 8 -30

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

n § § 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. Ch 8 -32

n § § 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 a. Ral Ch 8 -33