Discrete Mathematics Answers

Provide the answers to the following items. (5 items x 3 points) 

1. Given A = (2, 4, 6, 8} and B = {3, 4, 5, 6}, determine: 

a. A ∪ B b. A ∩ B 

2. Given A = {3, 5, 7, 9} and B = {4, 5, 6, 7}, determine: 

a. A - B 

b. B - A 

c. A ∩ B

PREDICATE LOGIC:

Write the following predicates symbolically and determine its truth value.

Note: Use at least three (3) values for the variables.

1. for every real number x, if x>1 then x – 1 > 1

2. for some real number x, x2 ≤ 0

C. Translate the following English sentence into symbol. (3 pts each)

1. No one in this class is wearing pants and a guitarist.

Let:

Domain of x is all persons

A(x): x is wearing pants

B(x): x is a guitarist

C(x): belongs to the class

Answer: __________________________________________________

2. No one in this class is wearing pants and a guitarist.

Let:

Domain of x is persons in this class

A(x): x is wearing pants

B(x): x is a guitarist

Answer: __________________________________________________

3. There is a student at your school who knows C++ but who doesn’t 

know Java.

Let:

Domain: all students at your school

C(x): x knows C++

J(x): x knows Java

Answer: ______________________________

PREDICATE LOGIC.(25 pts)

A. Let P(x) be the statement x

2 > x4. If the domain consists of the integers, 

what are the truth values? 

1. P(0) 

2. P(-1) 

3. P(1) 

4. P(2) 

5. ∃xP(x)

6. ∀xP(x)

Here are some scenarios:

• According to market research, a business has a 75% chance of making money in the first 3 years
• According to lab testing, of a certain kind of experimental light bulb will work after 3 years.
• According to experts, the likelihood of a car needing major repairs in the first 3 years is 0.7.

a. Write the scenarios in order of likelihood from least to greatest after three years: the business makes money, the light bulb still works, and the car needs major repairs.

Determine the truth value of each of these statements if the domain consists of all integers. NOTE: Explain how did you get the truth value on each statements in your own words.

a) ᴲxP(2n=3n)

b) ⱯxP(3n=<4n)

Use the table of propositional logical equivalences to show that ¬(p ∨

¬(p ∧)) is a contradiction.

Prove that (p ∧ q) → (p ∨ q) is a tautology using the table of propositional

equivalences.

If the solution of the recurrence relation αun−1 +βun−2 = f(n),(n ≥ 2) is

un = 1−2n+3.2

n

, then determine the values of α,β and f(n). (6)

b) A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at

the end of the year (after the interest is paid). Find a recurrence for the amount of

money after n years if you invest |2000. (4)

c) If 5 points are chosen in a square of side 2cm, show that there will always be two

points at a distance of at most √

2cm.

Write the following statements in symbolic forms.

a.It is cold and it is windy

b. If berries are ripe along the trail, hiking is safe if and only grizzly bears have not been seen in the area.

C. It is necessary to wash the boss's car to get promoted.

d. Winds from the south imply a spring thaw.

e.If you watch television,your mind will decay, and vice versa.

f.Low humidity and sunshine are sufficient for me to play tennis the afternoon.

g.It is snowing but, we will go out for a work.

Let the function g : Z → Z be defined by

g(x) = 7x + 3.

(i) Show that g is one-to-one.

(ii) Show that g is not onto.