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Example 1.What is the relative closure of the relationR= {(a, b) | a < b }

on the set of integers?

Solutions:The reflexive closure ofRis

R = {(a, b) | a < b } {(a, a) | a Z} = {(a, b) | a b}.

Example 2.What is the symmetric closure of the relationR = {(a, b) | a >

b}on the set of positive integers?

Solutions:The symmetric closure of R is the relation

R R-1= {(a, b) | a > b} {(b, a) | a > b} ={(a, b) | a b}

Example 3. LetRbe the relation on the set of all subway stops in New

York City that contains (a, b)if it is possible to travel from stop ato stop

bwithout changing trains What isRnwhen nis a positive integer? What

isR*?

Solution:The relationRncontains (a, b)if it is possible to travel from

stop ato stop bby making at most n 1changes of trains. The relation R*

consists of ordered pairs (a, b)where it is possible to travel from stop ato

stop bmaking as many changes of trains as necessary. (The reader should

verify these statements.)

Example 4. Show that the greatest than or equal relation ()is a partial

ordering on the set of integers.

Solutions:Since a a for every integer a, is reflexive. If a band b

a,then a = b. Hence, is antisymmetric. Finally, is transitive sincea

band bcimply that a c. It follows that is a partial ordering on theset of integers and (Z, )is a poset

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Example 5. The divisibility relation | is a partial ordering on the set of

positive integers, since it is reflexive, antisymmetric, and transitive, as

was shown in Section 6.1 We see that (Z+, |)is a poset (Z

+denotes the set

of positive integers.)

Example 6. Show that the inclusion relation is a partial ordering on the

power set of a set S.

Solutions:SinceA AwheneverAis a subnet of S, is reflexive. It is

antisymmetric sinceA BandB Aimply thatA = B. Finally is

transitive, sinceA BandB Cimply thatA C. Hence, is a partial

ordering onP(S), and (P(S), )is a poset

Example 7. In the poset (Z+, |),are the integers 3 and 9 comparable? Are

5 and 7 comparable?

Solutions: The integers 3 and 9 are comparable, since 3 | 9. The integers

5 and 7 are in comparable, because 5 is not divisible by 7 and 7 is not

divisible by 5

Example 8. The poset (Z, )is totally ordered, since a bor b a

whenever aand bare integers.

Example 9. The poset (Z+, |)is not totally ordered since is contains

elements that are in comparable, such as 5 and 7.

Example 10.Let Sbe a set. Determine whether there is a greatest

element and a least element in the poset (P(S), ).

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