Solved: a) Simplify the following Boolean expression using three variable maps F(x, y, z) = x y z + x’y’ z + x y’ z’ b)Simplify the following Boolean expression using four variable maps: F(w,x, y, z) = Σ(0,1,4,5,6,7,8,9)

an=-an-1+16an-2+4an-3-48an-4,where a°=0,a1=16,a2=-2,a3=142

On the basis of 5 years of data the following probability function for a company’s weekly demand for wool was: Amount of wool (kg) 2500 3500 4500 5500 Probability 0.35 0.45 0.20 0.05 What was the expected weekly demand for wool ba…

Consider Premises: If Claghorn has wide support, then he’ll be asked to run for the senate. If Claghorn yells “Eureka” in Iowa, he will not be asked to run for the senate. Claghorn yells “Eureka” in Iowa. Conclusion: Claghorn does not…

For any rational numbers r and s, 2r + 3s is rational.

Find all combinations of truth values for p, q and r for which the statement ¬p ↔ (q ∧ ¬(p → r)) is true

Suppose p is the statement 'There are 1,000 meters in one kilometer' and q is the statement 'You will give me a cake.' Select the correct symbolization for the statement 'There are 1,000 meters in one kilometer or you will not give me…

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

Use POLYA'S FOUR-STEP PROBLEM SOLVING STRATEGY to solve the problems. Layla is going to buy 30 multi-vitamin capsules, some P10 and some P18. If she has P340, what is the maximum number of P18 multi-vitamin capsules she can buy? (HI…

~(~p^q) ^(pvq) =p

P^~q implies r

Q.1.Let A={1 2 3 4 5} .Determine the truth value of each of the following statements justify your answers: (a) (EE x in A)(x+3=10) (c) (EE x in A)(x+3<5) (b) (AA x in A)(x+3<10) (d) o+x in A|(x+3<=7)

Let A = {2,4} and B = {6,8,10} and define relations R and S from A to B as follows: For all (x, y) E A ® B, XRy iff x|y, and xSy iff y - 4 = x. State explicitly which ordered pairs are in A x B, R, S, RUS, and RNS Expert's answer

Write a simple formula that generates the below mentioned terms. a) 1,2,2,3,4,4,5,6,6,7,8,8,... b) 1,10,11, 100, 101, 110,111,1000,1001,1010,1011

(b) What is the value of x after each of these statements is encountered in a computer program, if x =3 before the statement is reached? (i) if x +2=5 then x = 3*x +5 (ii) if (x +1=4) OR (2x +2=3) then x = x +1 (iii) if (2x +3=5)…

What is the value of x after each of these statements is encountered in a computer program, if x =3 before the statement is reached?

a) Determine whether are equivalent without using truth table.

A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at the end of the year (after the interest is paid). Find a recurrence for the amount of money after n years if you invest |2000.

If R, S and T are relations over the set A, then: Prove that If R⊆S, then T∘R ⊆ T∘S and R∘T ⊆ S∘T

We want to grade students by using IF or Nested if statement. The student name and marks are given. Write an IF or Nested IF statement to arrive at their grades i the grade column. The marks and the grades are as follows: Less than 6…

Let RR be a relation from the set AA to the set BB. Taking into account that R^{-1}=\{(b,a)\ |\ (a,b)\in R\}\subset B\times AR −1 ={(b,a) ∣ (a,b)∈R}⊂B×A, we conclude that Ran (R)=\{b\ |&b…

Let R be a relation from the set A to the set B, then: Prove that Ran (R)=Dom (R-1 ).

If R, S and T are relations over the set A, then: Prove that (S∩T)∘R= (S∘R)∩(T∘R).

Construct a formal proof of validity for the following argument. [10] 1. (N v O) → P 2. (P v Q) → R 3. Q v N 4. ¬ Q ∴ R

For propositions p, q, and r, determine whether 𝑝 ⟶ (𝑞 ⟶ 𝑟) and (𝑝 ⟶ 𝑞) ⟶ 𝑟 are logically equivalent.

Construct a formal proof of validity for the following argument. [10] 1. (N v O) → P 2. (P v Q) → R 3. Q v N 4. ¬ Q ∴ R

Use a Venn Diagram and shadings to find what X will be in terms of sets A en B and set operations so that : A or B = (A/B) or (A and B) or X With X disjoint To both A/B - A and B

Draw the digraph and the matrix of the relation R= {(1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3,4), (4, 1), (4, 2), (4, 3)} on the set A= {1, 2, 3, 4, 5}. Also decide whether it is reflexive, whether it is symmetric, whether it is ant…

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

Represent the following relation to a directed graph. relation R = {(1, 1), (1,2), (3, 2)} on set S = {1,2,3},}

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

Give a contrapositive proof of the theorem; "If n is an interfer and 3n + 2 is even, then n is even."?

a) Simplify the following Boolean expression using three variable maps F(x, y, z) = x y z + x’y’ z + x y’ z’ b)Simplify the following Boolean expression using four variable maps: F(w,x, y, z) = Σ(0,1,4,5,6,7,8,9)

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Question ID: mtid-5-stid-8-sqid-1265-qpid-1003