Discrete Mathematics Answers

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.” *

There are 60 science students tn a

Secondary School. 35 of whom study

Chemistry and 30 study Technical

Drawing, 12 out of those students atudy

Biology and chemistry but not Technical

Drawing, 10 study Chemistry but

neither biology nor Technical Drawing 11

study only Technical Drawing and

neither Biology nor Chemistry, 10 also

study Chemistry and Technical Drawing

only

a. How many mdents study all the three

subjects?

b. How many students study biology and

Technical Drawing but not Chemistry?

C. How many students study Biology only?

d How many students study biology all

together.

p: "I study for the test"

q: "I am sick"

r: "I fail the exam"

Translate the compound proposition below into words:

(¬p∨q)→r