Discrete Mathematics Answers

RULE OF INFERENCEe

A. What rule of inference is used in each of the following arguments?

Show solution. (5 pts each)

1. If I will read my modules, then I can answer all the activities. If I can

answer all the activities, then I will get high scores. Therefore, if I will

read my modules, then I will get high scores.

2. Rizza is an IT student. Therefore, Rizza is either an IT student or a

programmer

3. If it is national holiday, then school is closed. It is national holiday.

Therefore, the school is closed.

4. If Ann does not love numbers or if Ann does not love programming.

If Ann loves numbers, then she can be a mathematician. Therefore,

Ann can be a mathematician.

Let S = ℤ⁺, a ~ b if a − b is divisible by 2. Is S an equivalence relation?

Let p denote He is rich and let q denote He is happy. Write each statement in symbolic form using p and q. Note

that He is poor and He is unhappy are equivalent to ¬p and ¬q, respectively.

(a) If he is rich, then he is unhappy. (c) It is necessary to be poor in order to be happy.

(b) He is neither rich nor happy. (d) To be poor is to be unhappy

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

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.(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.