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. 

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. ∑1𝑖=−1 𝑆𝑖 (5 pts)

2. ∏ 𝑆𝑖 4𝑖=2 (5 pts)

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)

. 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. 

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.

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. ∑1𝑖=−1 𝑆𝑖 (5 pts)

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

Find for:

1. Elements of R

2. Domain and Range of R

3. Draw the digraph

4. Identify the properties of R

Consider the following statements. Write each statement into its symbolic form.

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

T1.2 Apply your controller established in Step 1 to the original nonlinear model (1), and validate if the

vehicle can still achieve any desired speed by simulations. Discuss the similarities and differences of

the responses between the linear model and nonlinear model under the same controller.

dx/dt=x-y dy/dt=x +3y

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.

III. 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.

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. ∑1𝑖=−1 𝑆𝑖

2. ∏ 𝑆𝑖 4𝑖=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.