8/13/2019 Solution Relation

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Solutions

1. a) Transitiveb) Reflexive, symmetric, transitive

c) Symmetric d) Antisymmetric

e) Reflexive, symmetric, antisymmetric, transitive f) None of these properties2. a) Being taller than is not reflexive(I am not taller than myself), nor

symmetric (I am taller than my duaghter, but she is not taller than I). It

is antisymmetric (vacuously, since we never haveAtaller thanB, and

Btaller thanA, even ifA=B). It is clearly transitive.

b) This is clearly reflexive, symmetric and transitive (It is an

equivalence relation See Section 6.5). It is not antisymmetric, since

twins, for example, are unequal people born on the same day.

c) This has exactly the same answers as part (b), since having the

same first name is just like having the same birthday.

d) This is clearly reflexive and symmetric. It is not antisymmetric,

since my cousin and I have a common gradparent, and I and mycousin have a common gradparent, but I am not equal to my cousin.

This relation is not transitive. My cousin and I have a common

gradparent; my cousinand her cousin on the other side of her family

have a common gradparent. My cousins cousin and I do not have a

common gradparent.

3. a) Symmetricb) Symmetric, transitivec) Symmetricd) Reflexive, symmetric, transitive

e) Reflexive, transitive

f) Reflexive, symmetric, transitive

g) Antisymmetric

h) Antisymmetric, transitive

4. (c), (d), (f).5. The relations in part (a),(b), and (e) are not asymmetric since they

contain pairs of the form (x, x). Clearly, the relation in part (c) is notasymmetric. The relation in part (f) is not asymmetric (both (1, 3)and

(3, 1)are in the relation). It is easy to see that the relation in part (d) is

asymmetric.

6. a) {(0, 0), (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), (3, 0), (3, 3)}

b) {(0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0)}

7. a) {(1, 1), (1,5), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 5), (5, 3),(5, 4)}

b) {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 5), (3, 1), (3, 3), (3, 4), (3, 5),

(4, 1), (4, 2), (4, 3), (4, 4), (5, 1), (5, 3), (5, 5)}

8/13/2019 Solution Relation

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c) {(1, 1), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2),

(3, 3), (3, 4), (3, 5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 1), (5, 2), (5, 3),

(5, 4), (5, 5)}

d) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 3), (2, 4), (2, 5), (3, 1),

(3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 1),(5, 2), (5, 3), (5, 4), (5, 5)}

e) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5)}

f) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5),(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5)}

8. a) {(1, 1), (1,2), (1, 4), (2, 2), (3, 3), (4, 1), (4, 2), (4, 4)}b) {(1, 1), (1,2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}

c) {(1, 1), (1,2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}

9. a) The maximal elements are the ones with no other elements abovethem, namely land m.

b) The minimal elements are the ones with no other elements belowthem, namely a, b,and c.

c) There is no greatest element, since neither lnor mis greater thanthe other

d) There is no least element, since neither anor bis less than the other

e) We need to find elements from which we can find downward pathsto all of a, b,and c. It is clear that k, l,and mare the elementsfitting this description.

f) Since kis less than both land m, it is the least upper bound of a, b,and c.

g)No element is less than bothfand h,so there are no lower bounds.h) Since there are no lower bounds, there can be no greatest lower

bound

10. a) 24, 45 b) 3, 5 c) No d) No

e) 15,45 f) 15 g) 15, 5,3 h) 15

11. a) {1, 2}, {1, 3, 4}, {2, 3, 4}b) {1}, {2}, {4} c) No d) No

e) {2, 4}, {2, 3, 4} f) {2, 4}

g) {3, 4}, {4} h) {3, 4}

12. a) yes b) no c) yes