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- 2. a) Consider the propositions โ2+3 = 5โ and โThe Sun rises in the Westโ. i) Write the disjunction of the statements and give its truth value. ii) Write the conjunction of the statements and give its truth value. iii) Write the exโฆ
- If the solution of the recurrence relation ฮฑunโ1 +ฮฒunโ2 = f(n),(n โฅ 2) is un = 1โ2n+3.2 n , then determine the values of ฮฑ,ฮฒ and f(n).
- 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 5 points are chosen in a square of side 2cm, show that there will always be two points at a distance of at most โ 2cm.
- A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?
- Write the expression x1 โจx2 โงx3 โจx4 in conjunction normal form and disjunctive normal form
- Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 1,โ2 and 3 with multiplicities 2,1 and 2, respectively. The relation also has a non-โฆ
- If a planar graph has the degree sequence (2,2,2,3,4,4,5), how many faces will it have? Draw a planar graph with this degree sequence and the number of faces obtained to check your answer
- How many numbers form 0 to 999 (0 and 999 inclusive) are indivisible by 7 or 11?
- From a survey of 120 people, the following data was obtained: 90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the threeโฆ
- 8. a) Find the generating function of the recurrence an = 6anโ1 โ5anโ2 +1 with initial conditions a0 = 2,a1 = 5.
- Express 3x 4 +4x 3 +2x 2 +x in terms of [x]4,[x]3,[x]2 and [x].
- Draw the Ferrar graph of the following partitions: i) 20 = 9+6+3+2 ii) 28 = 10+9+8+1 Also, write down the conjugate partitions of each of these partitions.
- Evaluate C74
- Not having the lecture is sufficient for Dr. Boateng to conduct a quiz or students to make presentations. If the students do not make presentations then Dr. Boateng will conduct the quiz. Therefore, whenever we have the lecture,โฆ
- Determine the validity of the following argument: Having a strong mathematical background is necessary and sufficient for understanding the concepts in Information Technology. Having a strong mathematical background and understandโฆ
- Prove that 2/n4-3 if and only off 4/n2+3
- Q1. a) Let ๐ = {๐ฅ: ๐ฅ โ ๐, 1 โค ๐ฅ โค 12}, ๐ด = {2๐ฅ: ๐ฅ โ ๐ ๐๐๐ ๐ฅ ๐๐ ๐ ๐๐ข๐๐ก๐๐๐๐ ๐๐ 4}, ๐ต = {๐ฅ: ๐ฅ โ ๐, ๐ฅ ๐๐๐ฃ๐๐๐๐ 2} ๐๐๐ ๐ถ = {๐ฅ: ๐ฅ โ ๐, ๐ฅ2 โค 16}. i) List the elements belong to the sets A, B and C respectively. (3 marks) ii) Find ๐ถ โ (๐ดฬ โฉโฆ
- Define a relation on the set S of all strings of letters: two strings are related if you can get one from the other by reversing one pair of adjacent letters. For example, cow ocw but cow woc.
- Let n be an integer. Use Definition 1.6 to explain why 2n + 9 is an odd integer 2n + 9 = 2( )+ 1
- In a class of 32 students, 18 offers chemistry, 16 offers Physics 22 offer Mathematics. 6 offer all three subjects, 3 offer Chemistry and Physics only and 5 offer only physics only. Each student offers at least one subject. Find the nโฆ
- If x=3 then x<2 is a statement or not?
- Identify if the following statements are predicate logic. Give a domain of discourse for each propositional function. A. The movie won the Academy Award for the past 2 years. B. 1 + 3 = 4 C. (x+2)2 is a prime number.
- Find the number of integers between 1 and 300 both inclusive that are divisible by the following types: (i). at-least one of 3, 5, 7. (ii). 3 and 5 but not 7. (iii). 5 but neither 3 nor 7
- For the following relation on the set {x:xโฮ and 1 โค x โค 12}. List (a) the ordered pairs belonging to the relation: (b) the transitive closure R = {(x, y): xy = 9}
- Suppose R(x,y) is the predicate โx understands y,โ the universe of discourse for x is the set of students in Discrete Class, and the universe of discourse for y is the set of examples in these lecture notes. Translate the following โฆ
- Let f: ZยฎZ be such that f(x) = x +1. Is f invertible? and if it is, what is its inverse?
- P(n): n! > 3n for n โฅ 7. What is the Base Step?
- Translate these statements into English sentence, where C(x) is "x is a comedian" and F(x) is "x is funny" and the domain consist of all people. a) Ax(C(x) -> F(x)) b) Ex(C(x) ^ F(x)) c) Ex(C(x) -> F(x))
- How many bit strings of length 14 contains the following types (I) atmost three is? (ii) an equal number of 0s and 1s (iii) an odd number of 1s
- Let P (x): x2/2 =x find the following and identify their truth values 1. (P) 1 2. (P) 2 3.โn, P(n) 4.โn, P(n)
- In a class of 35 students it is known that 24 of them do arts, 20 do chemistry and 22 do biology. All the students do at least one of the 3 subjects, 3 do all the three subjects while 7 do art and biology. 6 do art and chemistry but nโฆ
- Find the general solution of the recurrence relation: an = an-1 + 2an-2 , with a0 = 2 and a1 = 7.
- 1. Express the following system specifications using logical connectives: โThe message is scanned for viruses whenever the message was sent from an unknown system.โ โThe message was sent frโฆ
- cnf of q^(-q->(p^(-p->r)))
- 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
- 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
- For each of the arguments below, determine whether the argument is correct or incorrect and explain why. a) All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic. bโฆ
- Find and if for every positive integer , a) Ai={0,i} b) Ai=[-i,i].
- Use Karnaugh map to find a minimal sum for: E = xโyz + xโyzโt + yโzt + xyztโ + xyโzโt
- Use the Euclidean algorithm to find gcd(2074, 2457) = d.
- given that: P(n): n! > 3n for n โฅ 7. What is the Base Step?
- When A = 1, B = 0, C = 1. Then: F = C + C'B + BA' = 1
- Read and follow instructions.
- Let x be 5, x+y=10.
- DLSU-CCS is in Quezon City.
- Write these propositions using f and s and logical connectives. It is below freezing and snowing. It is below freezing, but not snowing. It is either below freezing, or snowing (or both). It is neither below freezing, nor snowing. Thโฆ
- State the converse, inverse, and contrapositive of each of the given statements. If it snows tonight, I will stay home. I go to the beach whenever it is a sunny summer day. A positive integer is prime only if it has no divisors otherโฆ
- IV. Use Logical Equivalence Rules to simplify the following. Construct the truth tables for each of the compound propositions and its simplified expression. ยฌbโง(aโb)โงa (aโb)โ(ยฌaโจb) ((pโq)โr)โง(ยฌqโ(pโงยฌq)) ยฌaโง(bโc)โง(ยฌbโจc)
- V. Determine whether each pair of propositions are logically equivalent or not. Use Logical Equivalence Rules. ยฌ(pโจ(ยฌpโงq)) and ยฌpโงยฌq ยฌ(pโq) and pโยฌq pโ(qโงr) and (pโq)โง(pโr) ยฌ(pโq) and pโq
- Determine if each statement is a proposition. If so, also determine its truth value. Read and follow instructions. Ok I answered the assignment. 2+3=5 Let x be 5, x+y=10. DLSU-CCS is in Quezon City.
- Formalize the following argument, then determine the result using logical Inferences Either Derek works or Avery does not work. If it is not true that both Avery works and Brandon does not work, then clearly Celina does not work. Hโฆ
- Consider the argument: "Mary is a diabetic. If Mary is a diabetic, then Frank is a television watcher. If Frank is a television watcher, then Mark is not unhappy. Either James is a watermelon, or Mark is happy." Formaliโฆ
- a) Show that the following logical equivalences hold for the Peirce arrowโ, where P โQ = ~ (P โจ Q). P โจ Q = (P โ Q) โ (P โ Q) P โง Q= (P โ P) โ (Q โ Q) b) Show that for the Shuffer stroke | P โง Q = (P | Q) | (P | Q) c) Use thโฆ
- Consider the argument: "Mary is a diabetic. If Mary is a diabetic, then Frank is a television watcher. If Frank is a television watcher, then Mark is not unhappy. Either James is a watermelon, or Mark is happy." Formaliโฆ
- Consider Premises: If there was a cricket match, then traveling was difficult. If they arrived on time, then traveling was not difficult. They arrived on time. Conclusion: There was no cricket match. Determine whether the conclusโฆ
- an=-an-1+16an-2+4an-3-48an-4,where aยฐ=0,a1=16,a2=-2,a3=142
- On the basis of 5 years of data the following probability function for a companyโs 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 baโฆ
- Consider Premises: If Claghorn has wide support, then heโll be asked to run for the senate. If Claghorn yells โEurekaโ in Iowa, he will not be asked to run for the senate. Claghorn yells โEurekaโ in Iowa. Conclusion: Claghorn does notโฆ
- 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 ยฌp โ (q โง ยฌ(p โ 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 meโฆ
- Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule if the integer (a โ 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? (HIโฆ
- ~(~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 ยฎ 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)โฆ
- 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.
- Assignment 1 Due: 6th April 2021 at 5.00 p.m. Total: 70 marks 1. Using the theorem divisibility, prove the following a) If a|b , then a|bc โa,b,cโโค ( 5 marks) b) If a|b and b|c , then a|c (5 marks) 2. Using any programming languโฆ
- 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.
- 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 RโS, then TโR โ TโS and RโT โ SโT
- 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 6โฆ
- 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 โ1 ={(b,a) โฃ (a,b)โR}โBรA, we conclude that Ran (R)=\{b\ |&bโฆ
- 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 (SโฉT)โR= (SโR)โฉ(TโR).
- Construct a formal proof of validity for the following argument. [10] 1. (N v O) โ P 2. (P v Q) โ R 3. Q v N 4. ยฌ Q โด R
- For propositions p, q, and r, determine whether ๐ โถ (๐ โถ ๐) and (๐ โถ ๐) โถ ๐ are logically equivalent.
- Construct a formal proof of validity for the following argument. [10] 1. (N v O) โ P 2. (P v Q) โ R 3. Q v N 4. ยฌ Q โด 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 antโฆ
- 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 yโฆ
- 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 โ pโฆ
- 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. โxP(x) 6. โxP(x) B. Write the following predicates symbolically and determine its truโฆ
- 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(โฆ
- 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 activitโฆ
- 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 programmiโฆ
- 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 raisedโฆ
- 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 โฆ
- 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. โฆ
- 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!&bโฆ
- 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 eaโฆ
- 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 eaโฆ
- 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. โxP(x) 6. โxP(x) B. Write the following predicates symboโฆ
- Using a Truth table, determine the value of the compound proposition ((๐ โจ ๐) โง (๏ฟข๐ โจ ๐)) โ (๐ โจ ๐).

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