MTH 221 Week 3
Chapter 7 Exercises:
7.1.5a) For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive.
R ⊆ Z+ x Z+ where a R b if a|b (read “a divides b,” as defined in Section 4.3)
The relation is reflexive, antisymmetric, and transitive
7.1.6) Which relations in Exercise 5 are partial orders? Which are equivalence relations?
The relation in part “a” is a partial order. The relations in “c” and “f” are equivalence relations.
7.2.2) If R is a reflexive relation on a set A, prove that R² is also reflexive on A.
Let x € A. R reflexive ===> (x, x) € R. (x, x) € R, (x, x) € R ===> (x, x) € R o R = R²
7.3.1) Draw the Hasse diagram for the poset (P(U), ⊆), where U = {1, 2, 3, 4}
Please see attached.
7.3.6a) For A = {a, b, c, d, e}, the Hasse diagram for the poset (A, R) is shown in Fig. 7.23.
determine the relation matrix for R.
(a) (b) (c) (d) (e)
(a) 1 1 1 1 1
(b) 0 1 0 1 1
M(R) = (c) 0 0 1 1 1
(d) 0 0 0 1 1
(e) 0 0 0 0 1
7.4.1a) Determine whether each of the following collections of sets is a partition for the given set A. If the collection is not a partition, explain why it fails to be.
A = {1, 2, 3, 4, 5, 6, 7, 8}; A1 = {4, 5, 6}, A2 = {1, 8}, A3 = {2, 3, 7}
The collection provides a partition of A
7.4.2a) Let A = {1, 2, 3, 4, 5, 6, 7, 8}. In how many ways can we partition A as A1 ∪ A2 ∪ A3 with
1, 2 ∈ A1, 3, 4 ∈ A2, and 5, 6, 7 ∈ A3?
There are 3 choices for placing 8 in A1, A2, or A3 so there are 3 partitions for A
Chapter 8 Exercises:
8.1.4) Annually, the 65 members of the maintenance staff sponsor a “Christmas in July” picnic for the 400 summer employees at their company. For these 65 people, 21 bring hot dogs, 35 bring fried chicken, 28 Bring salads, 32 bring desserts, 13 bring hot dogs and fried chicken, 10 bring hot dogs and salads, 9 bring hot dogs and desserts, 12 bring fried chicken and salads, 17 bring fried chicken and desserts, 14 bring salads and desserts, 4
7.1.5a) For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive.
R ⊆ Z+ x Z+ where a R b if a|b (read “a divides b,” as defined in Section 4.3)
The relation is reflexive, antisymmetric, and transitive
7.1.6) Which relations in Exercise 5 are partial orders? Which are equivalence relations?
The relation in part “a” is a partial order. The relations in “c” and “f” are equivalence relations.
7.2.2) If R is a reflexive relation on a set A, prove that R² is also reflexive on A.
Let x € A. R reflexive ===> (x, x) € R. (x, x) € R, (x, x) € R ===> (x, x) € R o R = R²
7.3.1) Draw the Hasse diagram for the poset (P(U), ⊆), where U = {1, 2, 3, 4}
Please see attached.
7.3.6a) For A = {a, b, c, d, e}, the Hasse diagram for the poset (A, R) is shown in Fig. 7.23.
determine the relation matrix for R.
(a) (b) (c) (d) (e)
(a) 1 1 1 1 1
(b) 0 1 0 1 1
M(R) = (c) 0 0 1 1 1
(d) 0 0 0 1 1
(e) 0 0 0 0 1
7.4.1a) Determine whether each of the following collections of sets is a partition for the given set A. If the collection is not a partition, explain why it fails to be.
A = {1, 2, 3, 4, 5, 6, 7, 8}; A1 = {4, 5, 6}, A2 = {1, 8}, A3 = {2, 3, 7}
The collection provides a partition of A
7.4.2a) Let A = {1, 2, 3, 4, 5, 6, 7, 8}. In how many ways can we partition A as A1 ∪ A2 ∪ A3 with
1, 2 ∈ A1, 3, 4 ∈ A2, and 5, 6, 7 ∈ A3?
There are 3 choices for placing 8 in A1, A2, or A3 so there are 3 partitions for A
Chapter 8 Exercises:
8.1.4) Annually, the 65 members of the maintenance staff sponsor a “Christmas in July” picnic for the 400 summer employees at their company. For these 65 people, 21 bring hot dogs, 35 bring fried chicken, 28 Bring salads, 32 bring desserts, 13 bring hot dogs and fried chicken, 10 bring hot dogs and salads, 9 bring hot dogs and desserts, 12 bring fried chicken and salads, 17 bring fried chicken and desserts, 14 bring salads and desserts, 4