Solved: 1. Determine whether each of the following statements is a proposition or not. If it is, give its truth value. • p: Mindanao is an island in the Philippines. • q: Find a number which divides your age. • r: My seatmate will get a perfect score in the Logic exam. • s: Welcome to the Philippines!

A function f is said to be one-to-one, or an injection, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. Note that a function f is one-to-one if and only if f(a) ≠ f(b) whenever a ≠ b. This way of expr…

∀n ≥ 1, Xn i=1 i(i!) = (n + 1)! − 1

The chairs of an auditorium are to be labeled with an uppercase English letter followed by a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently?

A set which contains all the possible element of the set under consideration is called ................................ Set

Use De Morgan's laws to find the negation of each of the following statements. a) Jan is rich and happy.

Give the truth value. P-~q,if q-~p is false

Let 𝐴={1,2,3,4}. Determine the truth value of each statement: i. ∀𝑥 ∈𝐴,𝑥+3<6

Consider the following English statements and express as a logical expression with the same meaning. The domain is the set of all students at Hogwarts. H(x): x is a member of the Hufflepuff. R(x): x is a member of Ravenclaw. …

1. Transform the following English statements into logical expressions. Identify the predicates, values, and their domain. Negate the logical expression and transform it back to English. a. The raptors are better than every o…

Let P (x, y) be a predicate. What is the negation of ∃x ∀y(P (x, y) ⇐⇒ P (y, x))?

Given the following three statements, what can you conclude about James’s ability to climb a mountain? Justify your answer. a. James is from Nova Scotia b. No one who loves lobster can climb a mountain. c. All people fr…

Consider the following predicates over the domain of all dogs: B(x): x barks W(x): x wags its tail S(x): x scratches the furniture T(x): x wants a treat. H(x): x goes to heaven Translate the following propositions from simple …

The federal Green Party recently elected a new leader: Bernie May. A party oﬃcial said 23,877 Green voters cast a ballot in this race. If the voters were partitioned into 12 regions based on population, give a proof by contradiction t…

Prove or disprove : for every real number x, |x − 8| + x ≥ 6.

Prove or disprove: The average of three real numbers is greater than or equal to at least one of the numbers.

What is the domain and range of the function f that assigns to each pair of negative integers x and y, the value x + y.

Write the following logical arguments as predicate expressions, defining the predicates used and domains of variables. For each argument, mention the inference rules used in each step. [6 marks] a) “Asim, a student in th…

Use the pigeonhole principle to give solutions to the following problems: (a) How many times must a single die be rolled to guarantee that some number is obtained at least twice? (b) How many times must two dice be rolled to guarant…

Express the following argument in symbolic form, and use the rules of inference and also MATLAB to show that it is logically valid. Ben is a student or a taxi driver. If Ben is a taxi driver, then he has a taxi licence. Ben has …

(a) In how many different ways can the letters of the word donkey be arranged? (b) In how many different ways can the letters of the word donkey be arranged if the letters wo must remain together (in this order)? (c) How many differ…

In an exam, a student is required to answer 10 out of 13 questions. Find the number of possible choices if the student must answer: (a) the first two questions; (b) the first or second question, but not both; (c) exactly 3 out of …

Let A = (8 -1 2 2 0 -5) B = (-1 7 3 -2 1 5) and C = (2 1 3 5) (a) Calculate AB and A + B if they exist. (b) Verify that (AB)C = A(BC). (c) Calculate C-1 A.

If 𝑈 = {1,2,3,4,5,6,7,8,9,10,11,12}, 𝐴 = {2,3,5,6}, 𝐵 = { 1,5,7,8,10,12}, 𝐶 = {8,10,11,12}. 𝐹𝑖𝑛𝑑 𝐴 ∪ 𝐵, 𝐴 ⊕ 𝐵, 𝐵 ∖ 𝐶, |𝑃(𝐴)|

Determine whether this statement about Fibonacci numbers is true. 2𝐹𝒙 − 𝐹𝒙−2 = 𝐹𝒙+1 for x> 1 where 𝐹𝒙 is the 𝒙𝑡ℎ Fibonacci number.

Write the following logical arguments as predicate expressions, defining the predicates used and domains of variables. For each argument, mention the inference rules used in each step. [6 marks] a) “Asim, a student in this class, ow…

A- For all integers a and b, if a + b is odd, then a is odd or b is odd. B- For any integer n the number (n3 - n) is even. C- Proof of De-Morgan’s Law

2.Determine the cardinality of each of the sets, A, B, and C defined below, and prove the cardinality of any set that you claim is countably infinite. A is the set of negative odd integers B is the set of positive integers less …

Use iteration, either forward or backward substitution, to solve the recurrence relation an=an−1−1 for any positive integer n, with initial condition, a0=1. Use mathematical induction to prove the solution you find is correct.

p ^ ~ (q ^ r)

Prove the identity laws in Table 1 by showing that a) A U 0 A. b) A n U = A.

Construct a truth tabal for each of these compound proposition

We call a positive integer perfect if it equals the sum of its positive divisors other than itself. (a) Prove that 6 and 28 are perfect numbers (b) Prove that if 2p − 1 is prime, then 2p−1

p ↔ ¬ (r˅q)

Asim, a student in this class, knows how to write programs in Java. Everyone who knows how to write programs in Java can get a high-paying job. Therefore, someone in this class can get a high-paying job

Show that inclusion relation ⊆ is a partial ordering on the power set of a set S. Draw the Hasse diagram for the partial ordering {(A,B) | A ⊆ B} on the power set P(S) where S = {a,b,c}.

Prove that the map f : C → R 2 , x + iy 7→ (x, y) is a bijection, i.e., a cardinal equivalence C ≈ R

Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Obtain the sum – of – products and product - of - sum canonical form for 𝑥1⨁(𝑥2 ⋆ 𝑥3 ′ )

Determine the cardinality of each of the sets, A, B, and C defined below, and prove the cardinality of any set that you claim is countably infinite. A is the set of negative odd integers B is the set of positive integers less th…

3. Let P(x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,” respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain cons…

Write the forpseudocode an algorithm that determines if a function is injective, assuming that you can iterate through all the elements in the domain and the co-domain in a finite number of steps. Recall the insertion sort algorithm …

Determine the cardinality of each of the sets, A, B, and C defined below, and prove the cardinality of any set that you claim is countably infinite. A is the set of negative odd integers B is the set of positive integers less th…

Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Using a truth table, prove or disprove the following: ~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

Show that the hexadecimal expansion of a positive integer can be obtained from its binary expansion by grouping together blocks of four binary digits, adding initial zeros if necessary, and translating each block of four binary digits…

Use properties of Boolean algebra to simplify the following Boolean expression (showing all the steps): (x + y′)(x′ + y′)′

Find the inverse of 35 modulo 11 by using extended Euclidean Algorithm

Construct a truth tabal for each of these compound proposition

Give the truth value. P-~q,if q-~p is false

Define a relation T from R to R as follows: For all real numbers x and y (X,y) E T means that y^2- x^2= 1. Is T a function? Explain

Determine whether each of these pairs of sets are equal. (1 pt each) a) {1,3,3,3,5,5,5,5,5}, {5,3,1} b) {{1}}, {1,{1}} c) ∅, {∅}

Let A = { 0,2,4,6,8,10 }, B = { 0,1,2,3,4,5,6 }, and C = { 4,5,6,7,8,9,10 }. (2 pts each) Find a) A ∩ B ∩ C b) A ∪ B ∪ C c) ( A ∪ B) ∩ C d) ( A ∩ B) ∪ C

Using a truth table, prove or disprove the following: ~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers. (1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) (2) ∀x((x < 0) ∨ (x^…

Rahul left his cycle in a parking lot that is filled with bikes and cycles, parked neatly in several rows. When he returns, he notices someone push all the vehicles on the left end of each row, causing them to fall. Rahul knows that…

In the game of poker, five cards are drawn from a deck of 52 cards. A set of 5 cards is said to be a hand. [A standard 52-card deck contains four suits of 13 cards each. The suits are spades (♠), clubs (♣), hearts (♥) and diamonds(♦…

Let R ⊆ S × S be an equivalence relation on a set S. For an element x ∈ S, let S(x) = {y ∈ S : (x, y) ∈ R}. Show that for every pair of elements x, y ∈ S, either S(x) = S(y) or S(x) ∩ S(y) = ∅.

Let A={a, b, c}, B={d, e}, C={a, d}. Find (i) A × B (ii) B × A (iii) A × (B C) (iv) (A C) x B (v) A ∆ (B – C) (vi) Complement of (A ∆ B) (vii) P(B C)

Find the singleton sets and the pair of disjoint sets from the following 1. A = {x |x and x4 − 4 =0} and B = {y | y and x4 − 16 =0 =0}.

Determine for which positive integer values of n, 3n^3+2≤n^4 and prove your claim by mathematical induction. I was able to determine n must be greater or equal to 4. But I can't seem to get through the inductive step.

Using the definition of "Big-O" determine if each of the following functions, f(x) = (xlogx)^2 - 4 and g(x) = 5x^5 are O(x^4) and prove your claims.

In how many ways can a set of five letters be selected from alphabets

For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers. (1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) (2) ∀x((x < 0) ∨ (x^…

Let R ⊆ S × S be an equivalence relation on a set S. For an element x ∈ S, let S(x) = {y ∈ S : (x, y) ∈ R}. Show that for every pair of elements x, y ∈ S, either S(x) = S(y) or S(x) ∩ S(y) = ∅.

Let A = {l, m, m, m, n, n, n, p, p, p} and B = {l, l, m, n, n, n, p, r, r}, find the operations of the following multisets: A. A + B B. A - B C. A U B D. A ∩ B

Using a truth table, prove or disprove the following: ~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

Consider the functions from the set of students in a discrete mathematics class. Under what conditions is the function one-to-one if it assigns to a student his or her a) mobile phone …

Suppose that p and q are any statements. By constructing the truth tables, show that the statement ¬ (p V q) & (¬ p) ∧ (¬ q) are logically equivalent.

Define the following terms using mathematical notations. Also provide example for each term- Subset Universal set Power set

Show the following operations on set(s)- Complement of a set Symmetric difference of two sets

Use mathematical induction to prove the following: An = (1 3n − 1 0 3n) for all integers n ≥ 1, where A = (1 2 0 3) After you have proved the above statement using mathematical induction, write a MATLAB program to veri…

Which is the most suitable graph representation scheme for a dense graph? Draw its representation with the help of an example. What is the space complexity of a such graph representation scheme?

Suppose T(n) and f(n) and two functions. Write asymptotic notations (Ο, Ω, Θ) using these two functions and explain the growth rate of these functions in each notation.

20. Give an example of a function from N to N that is a) one-to-one but not onto. b) onto but not one-to-one. c) both onto and one-to-one (but different from the iden- tity function). d) neither one-to-one nor onto.

Identify whether the given path in the graph is (a) A simple path (b) A cycle (c) A simple cycle (i) (b, b) (ii) (a, d, c, d, e) (iii) (e, d, c, b) (iv) (d, c, b, e, d) (v) (a, d, c, b, e)

Draw a graph having the given properties (a) Four vertices each of degree 1 (b) Six vertices each of degree 3

Determine the truth value of each of these statements if the domain for all variables consists of all integers. a) ∀n(n2 ≥ 0) b) ∃n(n2 = 2) c) ∀n(n2 ≥ n) d) ∃n(n2 < 0)

Given three sets A, B, and C. Suppose we know that the union of the three sets has cardinality 182. Further, |A| = 92, |B| = 41, |C| = 118. Also, |A ∩ B| = 15, |A ∩ C| = 42, and |A ∩ B ∩ C| = 10. Find |B ∩ C|.

Show that (p → q) → (r → s) and (p → r) → (q → s) are not logically equivalent. The dual of a compound proposition that contains only the logical operators ∨, ∧, and ¬ is the compound proposition obtained by replacing each ∨ by ∧,…

Given three sets A, B, and C. Suppose we know that the union of the three sets has cardinality 182. Further, |A| = 92, |B| = 41, |C| = 118. Also, |A ∩ B| = 15, |A ∩ C| = 42, and |A ∩ B ∩ C| = 10. Find |B ∩ C|.

Given three sets A, B, and C. Suppose the the union of the three sets has cardnality 280. Suppose also that |A| = 100, |B| = 200, and |C| = 150. And suppose we also know |A∩B| = 50, |A∩C| = 80, and |B∩C| = 90. Find the cardinality of …

Out of 300 students taking discrete mathematics 60 take coffee , 27 take coco, 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and coco, 3 take chocolate tea and coco, 20 take coco only, 2 take tea, chocolate a…

{(1,2), (3,5)} is a partition of {1,2,3,4,5}?

1.) {(1,2), (3,5)} is a partition of {1,2,3,4,5}? let s={1,2,3}. in each case give an example of relation r on s that has the stated properties 2.) r is not symmentric, not reflexive and not transitive. 3.) r is transitive and re…

1. Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends. 2. Show that if there are 30 …

C. Given A= {d}, B= {c, d}, C= {a, b, c}, D = {a, b}, E= {a, b, d}. Determine whether the following statements are True or False. 10. 𝐴 ⊂ 𝐵 11. 𝐶~𝐷 12. 𝐷 ⊈ 𝐸 13. n(C) =3 14. C and D are joint sets 15. E has 16 subsets D…

6-7. A = {the set of the first 5 positive integers divisible by 10} 8-9 C = {Cavite, Laguna, Batangas, Rizal, Quezon}

B.Write the following sets using rule method: 6-7. Write A= {10, 20, 30, 40, 50} using rule method. 8-9. Write C={𝑥|𝑥 𝑖𝑠 𝑎 province in the region of CALABARZON} using roster method

Suppose T(n) and f(n) and two functions. Write asymptotic notations (Ο, Ω, Θ) using these two functions and explain the growth rate of these functions in each notation.

Using a truth table, prove or disprove the following: ~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

List all strings over X = {0,1} of length 2 or less.

Show that A = B where A = {x│x² - 4x +4 =1}, B = {1,3}

Let 𝑄(𝑥, 𝑦) denote "𝑥 + 𝑦 = 𝑦“. What are the truth values of the quantifications ∃𝑦∀𝑥𝑄(𝑥, 𝑦) and ∀𝑥∃𝑦𝑄(𝑥, 𝑦) where the domain for all variables consists of all real numbers?

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Question ID: mtid-5-stid-8-sqid-3354-qpid-2053