Discrete Mathematics Answers

Use the Euclidean algorithm to find integers 𝑥 and 𝑦 such that 2640𝑥 + 2110𝑦 = 10

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

solution. (5 pts each)

1. If it will rain today, then the classes are suspended. The classes are not

suspended today. Therefore, it did not rain today.

2. If you read your module today, then you will not play ML today. If you

cannot play ML today, you can play ML tomorrow. Therefore, you read

your module today, then you will play ML tomorrow. 

B. Write each statement into its symbolic form.(3 pts each)

Let

x: PJ is a mathematician

y: MJ is a programmer

a. PJ is not a mathematician.

b. PJ is a mathematician while MJ is a programmer.

c. If PJ is a mathematician then MJ is not a programmer.

d. PJ is a mathematician or if PJ is a mathematician then MJ is a

programmer.

e. Either PJ is a mathematician and MJ is a programmer, or neither PJ is

a mathematician nor MJ is a programmer. 

C. Show whether or not p → q ≡ (p ^ q) v (𝒑

̅ ^ 𝒒̅) (10 pts)

D. Let P(x) denote the statement 𝟏

𝒙

𝟐+𝟏

> 1. If its domain are all real numbers,

what is the truth value of the following quantified statement? (5 pts each)

1. ∃xP(x)

2. ∀xP(x)

A. Determine whether each of the following propositions is true or false. Write

T if it is true and F if it false.

1. Five is a prime number.

2. City of Ilagan is the capital of Isabela.

3. The world is flat.

4. Any number raised to zero is always equal to one.

5. 2 + 4 = 7

6. 4 < 7 or 3 < 2

7. If 9 is not prime then 2 is prime.

8. – 3 > - 2

9. 0.1 > ½

10. {-2, -1, 0, 1, 2, 3} ∩ {0, 1, 3, 4} = {1, 2, 3}

RULE OF INFERENCE:

B. Determine if the following argument is valid. Explain by using rule of

inference. (5 pts each)

1. If you perform every programming problem in the module, then you

will learn programming. You learned programming. Therefore, you

perform every programming problem in the module.

2. Not everyone likes to go to the hospital; hence, there is someone

who does not like to go to the hospital.

III.

RULE OF INFERENCE (30 pts)

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. 

PREDICATE LOGIC:

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: _______

II.

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)

B. Write the following predicates symbolically and determine its truth value.

Note: Use at least three (3) values for the variables. (5 pts each)

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

2. for some real number x, x2 ≤ 0

B.Let p, q and r denotes the following statements:

p: A square has four equal side

q: Rectangle has 2 parallel sides

r: A square is a rectangle. 

1. Express each of the following into English sentence. (3 pts each)

a. r ^ q → p

b. p

̅ → q

c. q → p

̅ v r

2. Write T if the above item is true and F if it false. Show solution. (3 pts

each)

a.

b.

c.

C. Show whether or not p ↔ q ≡ (p → q) ^ (q → p) (8 pts)

D. Find the converse, inverse and contrapositive of the implication: “If today is

Monday then, I have an exam today.” (3 pts each)

1. Inverse:

2. Converse:

3. Contrapositive: 

PROPOSITIONAL LOGIC:

A. Which of the following are propositions? Write P for proposition and NP for

not proposition.

1. Why should we study Discrete Mathematics?

2. 10 is divisible by 3

3. 3>1 and 4 is a not an integer

4. Wear your facemask.

5. Fifteen is an even number.

6. Go directly to him.

7. If 8 is prime, then 56 is not prime.

8. From where are you?

9. Some prime is even.

10. A + B is equal to C