Search & Filtering
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.
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.
