Discrete Mathematics Answers

2. Determine the truth value of each of these statements if the domain consist of all integers.

A. ∀n (n+1>n)

B. Ǝn (n = -n)

C. Ǝn (2n = 3n)

D. ∀n (3n ≤ 4n)

1. Let Q(x) denote the statement “x is an integer”. What are the truth values?

A. Q(-1)

B. Q(0)

C. Q(8/2)

D. Q(sqrt(-4))

E. Q(sqrt(4))

 Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output ((¬p ¬r)𝖠¬q) (¬p 𝖠 (q ∨ r)) from input bits p, q, and r.

Given g(x) = (x + 1)(x^2 − x), g: R → R where R is the set of real numbers.

a) Find the domain and range of the function g. (2 marks)

b) Determine whether the function is injective, surjective, and/or bijective. Justify

your answers. (7 marks)

Let Q and R be any two sets given, prove that [Q̅ ∪ (Q − R)][overlined] = Q ∩ R.

2. Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output

((￢p ∨￢r) ∧￢q) ∨ (￢p ∧ (q ∨ r)) from input bits p, q, and r.

Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output

(p ∧￢r) ∨ (￢q ∧ r) from input bits p, q, and r.

1. a. Construct a truth table for (p ↔ q) and (p → q) ^ (q → p).

b. Determine whether these compound propositions are logically equivalent.

2. Let R(x, y): x² + y² = 1. Find the truth values of the propositions R(2 3 , 6 4 ) and R(3 5 , 7 4 ) .

3. Let P(x, y, z): x + y = z, where x, y and z are all real numbers.

a. Express the quantifications ∀x∀y∃z P(x, y, z) and ∃z∀x∀yP(x, y, z) as statements.

b. Find the truth value of the quantifications ∀x∀y∃z P(x, y, z) and ∃z∀x∀yP(x, y, z).

c. Determine whether both quantifications are logically equivalent.

Translate the given statement into logical expression using the propositions provided. "To use the wireless network in the airport you must pay the daily fee unless you are a subscriber to the service." Express your answer in terms of a: “You can use the wireless network in the airport,” d: “You pay the daily fee,” and s: “You are a subscriber to the service.” *