8/13/2019 Exercise Relation

1/3

1

Exercise

1. For each of the following relations on the set {1, 2, 3, 4}, decidewhether it is reflexive, whether it is symmetric, whether it is

antisymmetric, and whether it is transitive.a) {(2, 2),(2, 3), (2 ,4), (3, 2), (3, 3), (3, 4)}

b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}c) {(2, 4), (4, 2)}d) {(1, 2), (2, 3), (3, 4)}e) {(1, 1), (2, 2), (3, 3), (4, 4)}f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}

2. Determine whether the relationRon the set of all people is reflexive,symmetric, antisymmetric, and/or transitive, where (a, b) R if and

only ifa) ais taller than b.b) aand bwere born on the same day.c) ahas the same first name as b.d) aand bhave a common grandparent.

3. Determine whether the relationRon the set of all integers is reflexive,symmetric, antisymmetric, and/or transitive, where (x, y) R if and

only if

a)x y.b)

xy

1.c)x = y + 1orx = y 1.

d)x y (mod 7).e)xis a multiple ofy.f) xandyare both negative or both nonnegative.g)x = y2. h)x y2.

- A relationRon the set ofAis irreflexiveif for every a A, (a, a) R. That is,Ris irreflexive if no element InAis related to itself.

4. Which relations in Exercise 1 are irreflexive?- A relationRis called asymmetricif (a, b) Rimplies that (b, a) R.5. Which relations in Exercise 1 are asymmetric?6. LetRbe the relation on the set {0, 1, 2, 3}containing the ordered pairs

(0, 1), (1, 1), (1, 2), (2, 0), (2, 2),and (3, 0).Find the

a) reflexive closure ofR.b) symmetric closure ofR.

8/13/2019 Exercise Relation

2/3

2

7. Let Rbe the relation on the set {1, 2, 3, 4, 5}containing the orderedpairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2),and (5, 4).Find

a)R2. b)R

3.

c)R4. d)R5.

e)R6

. f) R*

.8. Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3),

(4, 1)} that is

a) reflexive and transitive.b) symmetric and transitive.c) reflexive, symmetric, and transitive.

9. Answer the following questions for the partial order represented bythe following Hasse diagram.

a) Find the maximal elements.b) Find the minimal elements.c) Is there a greatest element?d) Is there a least element?e) Find all upper bound of {a, b, c}.f) Find the least upper bound of {a, b, c},if it exists.g)

Find all lower bound of {f, g, h}.h) Find the greatest lower bound of {f, g, h},if it exists.

8/13/2019 Exercise Relation

3/3

3

10.Answer the following questions concerning the poset ({3, 5, 9, 15, 24,45}, |).

a) Find the maximal elements.b) Find the minimal elements.c) Is there a greatest element?d) Is there a least element?e) Find all upper bounds of {3, 5}.f) Find the least upper bound of {3, 5},if it exists.

g) Find all lower bounds of {15, 45}.h) Find the greatest lower bound of {15, 45},if it exists.

11.Answer the following questions concerning the poset ({{1}, {2}, {4},{1, 2}, {1, 4}, {2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}} ).

a) Find the maximal elements.b)

Find the minimal elements.c) Is there a greatest element?

d) Is there a least element?e) Find all upper bounds of {{2}, {4}}.f) Find the least upper bound of {{2}, {4}},if it exists.g) Find all lower bounds of {{1, 3, 4}, {2, 3, 4}}.h) Find the greatest lower bound of {{1, 3, 4}, {2, 3, 4}},if it exists.

12.Determine whether the posets with the following Hasse diagrams arelactices.