Deck 9: Relations

Question 1

Show that the inclusion relation, , is a partial ordering on the set of all subsets of

Answer 1

We see that set inclusion is reflexive since A A whenever A is a subset of . Since A B and B A imply that A=B whenever A and B are subsets of , we see that set inclusion is antisymmetric. Now suppose that A B and B C where A, B , and C are subsets of Then A C , so set inclusion is transitive.

Question 2

Show that the relation is an equivalence relation on the set of rational numbers. What are the equivalence classes of 0 and 12 ?

Answer 2

Question 3

Consider the poset with the following Hasse diagram. (a) Find all maximal elements of the poset. (b) Find all minimal elements of the poset. (c) Find the least element of the poset if it exists, or show that it does not exist. (d) Find the greatest element of the poset if it exists, or show that it does not exist. (e) What is the greatest lower bound of the set {a, b, c}? (f) What is the least upper bound of the set {a, b, c}?

Answer 3

Question 4

What is the transitive closure of the relation in problem 3?

Question 5

Which ordered pairs are in the relation { (x, y) | x > y + 1 } on the set {1, 2, 3, 4}?

Question 6

Suppose that R1 and R2 are symmetric relations on a set A. Prove or disprove that R1 − R2 is also symmetric.

Question 7

Find the reflexive closure and the symmetric closure of the relation {(1,2),(1,4),(2,3),(3,1),(4,2)} on the set {1,2,3,4} .

Question 8

(a) Show that the relation and are bit strings containing the same number of 0s } is an equivalence relation.
(b) What are the equivalence classes of the bit strings 1, 00, and 101 under the relation ?

Question 9

What is the join of the 3-ary relation {(Lewis, MS410, N507), (Rosen, CS540, N525), (Smith, CS518, N504), (Smith, MS410, N510)} and the 4-ary {(MS410, N507,Monday, 6: 00), (MS410, N507, Wednesday, 6: 00), (CS540, N525, Monday, 7: 30), (CS518, N504, Tuesday, 6: 00), (CS518, N504, Thursday, 6: 00)} with respect to the last two fields of the first relation and the first two fields of the second relation?

Question 10

(a) Are the sets {1,3,6},{2,4,7} , and {5} a partition of {1,2,3,4,5,6,7} ?
(b) Are the sets {1,2,4,5},{3,6,7} , and {2,3} a partition of {1,2,3,4,5,6,7} ?

Question 11

What are the minimal and maximal elements in the poset with the following Hasse diagram? Are there least

Question 12

Consider the following relations on the set of positive integers. (a) Which of these relations are reflexive? Justify your answers. (b) Which of these relations are symmetric? Justify your answers. (c) Which of these relations are antisymmetric? Justify your answers. (d) Which of these relations are transitive? Justify your answers.

Question 13

b) and (b,
c) with
a = b and b = c. R1 is not transitive since (3, 1) and (1, 3) belong to R1 but (3, 3) is not in R1. R3
is not transitive since (1, 2) and (2, 1) belong to R3 but (1, 1) does not belong to R3.