Solved: Let R5 = {(1,2), (1,4), (1,6), (2,2), (2,4), (2, 6), (3,2), (3,4), (3,6), (4,2), (4,4), (4,6), (5,2), (5,4), (5,6)} and R6 = {(1,2), (1,4), (1,6), (2,4), (2,6), (3,6)} be relations. How many ordered pairs are there in R5 U R6?

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 y…

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…

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 tru…

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(…

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 activit…

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 programmi…

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…

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 …

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. …

Let's say we have 63 balls in the urn, and need to choose 6 balls. Six balls have 6! orders, thus the number of choices for choosing 6 balls out of 63 without any repetition such that the order does not count is \dfrac{63!}{57!&b…

An urn contains numbered balls. How many ways can we choose balls out of the urn (without repetition, the order does not count)6 63

PROPOSITIONAL LOGIC: 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 ea…

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 symbo…

Using a Truth table, determine the value of the compound proposition ((𝑝 ∨ 𝑞) ∧ (￢𝑝 ∨ 𝑟)) → (𝑞 ∨ 𝑟).

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 …

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 cl…

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…

RULE OF INFERENCE A. What rule of inference is used in each of the following arguments? Show solution. 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 sc…

RULE OF INFERENCE B. Determine if the following argument is valid. Explain by using rule of inference. 1. If you perform every programming problem in the module, then you will learn programming. You learned programming. Therefor…

Let the function g : Z → Z be defined by g(x) = 7x + 3. (i) Show that g is one-to-one. (ii) Show that g is not onto.

Write the following statements in symbolic forms. a.It is cold and it is windy b. If berries are ripe along the trail, hiking is safe if and only grizzly bears have not been seen in the area. C. It is necessary to wash the boss's c…

If the solution of the recurrence relation αun−1 +βun−2 = f(n),(n ≥ 2) is un = 1−2n+3.2 n , then determine the values of α,β and f(n). (6) b) A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at the e…

Prove that (p ∧ q) → (p ∨ q) is a tautology using the table of propositional equivalences.

Use the table of propositional logical equivalences to show that ¬(p ∨ ¬(p ∧)) is a contradiction.

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)

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 expe…

PREDICATE LOGIC: Write the following predicates symbolically and determine its truth 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…

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. …

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

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) …

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

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. The…

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 per…

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?

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 …

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)

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 …

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 programme…

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. ∀x…

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…

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 …

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…

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…

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. 𝑃𝑜…

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

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 t…

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…

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…

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 𝑥…

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)…

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)

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 sour…

MATHEMATICAL INDUCTION AND RECURRENCE Solve the following. (10 pts each) 1. Prove P(n) = n2 (n + 1) 2. Recurrence relation an = 2n with the initial term a1 = 2.

MATHEMATICAL INDUCTION AND RECURRENCE 1. What is the base case for inequality 3n > n2 , where n = 2? (2 pts) A. 3 > 1 B. 9 > 4 C. 6 > 4 D. 4 < 9 2. For the mathematical induction to be true, what type of number should be the v…

MATHEMATICAL INDUCTION AND RECURRENCE 5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? (2 pts) A. (k + 1)2 (k + 2)(k) B. (k + 1)2 (k + 2)(k) C. (k + 1)(k + 3)(k) D. (k + 1)2 (k + 3)(k) 6. Using the principle of m…

p → ~q

Suppose you have a graph with v vertices and e edges that satisfies v=e+1. Must the graph be a tree? Prove your answer.

Proof by contradiction that if n is a positive integer, then n is odd if and only if 5n + 6 is odd

Show that if x is an integer then x2+x-41= 0 produce prime numbers.

As you now that 03 0 13 1 23 8 33 27 43 64 53 125 so, we have a proof that every positive integer is the sum of the cubes of eight non negative integers. For example 7= 13 + 13 + 13 +13+13+13+ 13 +03 12= 23+ 13 + 13 + 13 +13+…

Show that if x is an integer then x2+x-41= 0 produce prime numbers

Using propositional logic, verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Use the truth table to transform each of the following wffs into the full conjunctive normal form (P→Q)→P. P→(Q→P) (P∨Q)∧R. P→Q∧R. Q∧¬P→P

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RULE OF INFERENCE 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…

What rule of inference is used in each of the following arguments? Show solution. 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 m…

Draw the Hasse diagram for divisibility on the set, {1, 2, 3, 4, 5, 6, 7, 8}?

using truth tables, verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

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verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a b R, ) if the integer product of (ab) is divisible by 4. List the elements of R and its inverse?

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Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where: 𝑅 = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has:…

Give an indirect proof of the theorem; “If 𝑛 is an integer and n3 +13is odd, then n is even.”?

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In the following argument, determine the validity or otherwise of the Statement: a) If you aren’t polite, you won’t be treated with respect. You aren’t treated with respect. Therefore, you aren’t polite.

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Draw the Hasse Diagram representing the partial ordering {(a, b) | a divides b} on {1, 2, 3, 4, 5, 6, 10, 15, 20, 30, 60}?

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Let R be the relation on the set A {1,2,3,4,5,6,7} defined by the rule (a,b) elements of R if the integer product of (a,b) is divisible by 4. List the elements of R and its inverse?

Express the following using language of Predicate Calculus, where it is understood that the people being discussed are in the courtroom. If any sentence is ambiguous, give all symbolic versions. (i) All judges are sober (ii) There is …

Let f be a function from the set A to the set B. Let S and T be two disjoint subsets of A (i.e S∩T=∅); then which of the following cannot be true: a) if f is invertible, then f(S)∩f(T)=∅ b) if f(S)∩f(T)≠∅, then f is a one-to-one…

Let R5 = {(1,2), (1,4), (1,6), (2,2), (2,4), (2, 6), (3,2), (3,4), (3,6), (4,2), (4,4), (4,6), (5,2), (5,4), (5,6)} and R6 = {(1,2), (1,4), (1,6), (2,4), (2,6), (3,6)} be relations. How many ordered pairs are there in R5 U R6?

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Question ID: mtid-5-stid-8-sqid-1294-qpid-1032