Discrete Mathematics Answers

E. PREDICATE LOGIC. Rewrite each sentence symbolically and determine the truth values. Write T if it is 

true and F if it is false. Show complete solution. (5 pts each)

1. For some integer x, 𝑥 = 𝑥2 − 2

2. For every real number x, 𝑖𝑓 𝑥2 − 1 > 𝑥 𝑡ℎ𝑒𝑛 𝑥 + 1 > 1

3. For some integer n, 4n = 3n + 1

F. RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is 

used in each of the following arguments? Show solution. (4 pts each)

1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the 

problem was solved. If Jen wrote a Java source code, then the problem was solved. 

2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be 

vaccinated.

G

. PROBLEM SOLVING. 

A. SET. Let A, B and C are sets and U be universal set. 

U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e}

A = {-1, 1, 2, 4}

B = {0, 2, 4, 6}

C = {b, c, d}

Find for the following. Show complete solutions. (3 pts each)

1. 𝐵 ∪ 𝐶

2. 𝐴 − 𝐵 𝑥 𝐶

3. 𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡 𝑜𝑓 𝐶

4. |𝑃(𝐵)|

B. SEQUENCES. Consider the sequence {Sn} defined by Sn = 2n – 5, where 𝒏 ≥ −𝟏. 

Find for:

1. ∑𝑆𝑖1𝑖=−1

2. ∏𝑆𝑖4𝑖=2

C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x ≥ y.

Find for:

1. Elements of R (3 pts)

2. Domain and Range of R (2 pts)

3. Draw the digraph (3 pts)

4. Identify the properties of R (2pts)

 Consider the following statements. Write each statement into its symbolic form.(2 pts each)

Let:

p: Ann solved the problem in discrete math.

q: Bryan solved the problem in discrete math.

r: Cris solved the problem in discrete math.

s: Derynn solved the problem in discrete math.

1. If Derynn solved the problem in discrete math then Bryan and Cris solved it too.

2. Cris solved the problem in discrete math only if Ann and Bryan didn’t solve.

3. Derynn solved the problem in discrete mathematics if and only if Cris solved it and Ann doesn’t solved

4. If Derynn solved the problem in discrete mathematics, then if Cris doesn’t solve it then Ann solved it.

5. Cris solved the problem in discrete mathematics provided that Derynn doesn’t solved, but if Derynn solved it, then 

Bryan doesn’t solve it.

A. Show whether or not p → q ≡ (p ^ q) v (𝒑̅ ^ 𝒒̅) 

B.Let P(x) denote the statement

1

------

x2+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)

C. 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.

A. Write each statement into its symbolic form

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. 

PREDICATE LOGIC:

A.Write the following predicates symbolically and determine their true 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 

B. Translate the following English sentence into a 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:

The 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)

PROPOSITIONAL LOGIC:

A. 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.

a. r ^ q → p

b. p̅ → q

c. q → p̅ v r

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

each)

a.

b.

c

B. Show whether or not p ↔ q ≡ (p → q) ^ (q → p)

C. 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: 

What is the symbolic form of the statement, “No one in this class is wearing pants and a guitarist” if the domain of x is persons in this class, A(x): x is wearing pants and B(x): x is a guitarist? 

RULE OF INFERENCE:

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