Discrete Mathematics Answers

4) Express the negations of each of these statements so that all negation symbols immediately precede predicates.

a) Rita will move to Oregon or Washington:

b) if it rains today then the home team wins:

c) ∃x∃yP(x, y) ∧ ∀x∀yQ(x, y).

d) ∃x∃y(Q(x, y) ↔ Q(y, x)).

38)

Let 𝐴 = {2,3,4,5,6,7,8,9} and define an order relation ∼ on 𝐴 by the following: For all 𝑎, 𝑏 ∈ 𝐴, 𝑎 ∼

𝑏 if, and only if, there exists an integer 𝑡 such that 𝑏 = 𝑡𝑎. Then, in the poset (𝐴, ∼)

A) The minimal elements are 2 and 3 and the maximal elements are 6, 8 and 9.

B) The minimal elements are 2, 3, 4 and 5 and the maximal elements are 6, 7, 8 and 9.

C) There are no minimal elements, nor any maximal elements.

D) The minimal elements are 2, 3, 5 and 7 and the maximal elements are 5, 6, 7, 8 and 9.

39)

Let 𝐴, 𝐵, 𝐶 and 𝐷 be sets, and let 𝑅 ⊆ 𝐴 × 𝐵, 𝑄 ⊆ 𝐵 × 𝐶 and 𝑃 ⊆ 𝐶 × 𝐷 be relations. Then the

relations (𝑅 ∘ 𝑄) ∘ 𝑃 and 𝑅 ∘ (𝑄 ∘ 𝑃), from 𝐴 to 𝐷

A) must not be equal.

B) are sometimes equal, and sometimes not, depending on the sets 𝐴, 𝐵, 𝐶,𝐷 and the relations 𝑅, 𝑄

and 𝑃.

C) must be equal.

D) are not defined

In a Hasse diagram of a poset (𝐴, ≤), a pair of distinct elements 𝑎 and 𝑏 of 𝐴 are connected by a line

segment if, and only if,

A) either 𝑎 ≤ 𝑏 and there is no 𝑐 ∈ 𝐴 such that 𝑎 ≤ 𝑐 ≤ 𝑏, or else 𝑏 ≤ 𝑎 and there is no 𝑐 ∈ 𝐴 such

that 𝑏 ≤ 𝑐 ≤ 𝑎.

B) either 𝑎 ≤ 𝑏 or else 𝑏 ≤ 𝑎.

C) 𝑎 ≰ 𝑏 and 𝑏 ≰ 𝑎.

D) 𝑎 ≤ 𝑏.

Much appreciate!

4) Express the negations of each of these statements so that all negation symbols

immediately precede predicates.

a) Rita will move to Oregon or Washington.

b) if it rains today then the home team wins.

c) ∃x∃yP(x, y) ∧ ∀x∀yQ(x, y).

d) ∃x∃y(Q(x, y) ↔ Q(y, x)).