Solved: I. SET. A. List the members of these sets 1. {x | x is a positive integer less than 10} 2. {x | x is an integer such that x2 = 2} B. Consider the following sets. Universal set (U) = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d} A = {a, b, c} B = {-2, -1, 0, 1, 2} C = {2, 4, 6} Find for the following: a. A U B b. 𝐴 ∩ 𝐵 c. B – C d. C x A e. 𝐵 ∩ f. 𝐴̅ − 𝐵 g. C – (𝐵̅ − 𝐴) h. i. P (A) j. |𝑃(𝐵)|

Use the Well Ordering Principle to prove that any integer greater than or equal to 23 can be represented as the sum of nonnegative integer multiples of 6, 7 and 17.

Let α, β be roots of the equation x^2− 3x − 1 = 0. For each nonnegative integer n, let y_n = α^n + β^n . Show that gcd(y_n, y_(n+1)) = 1 for each nonnegative integer n.

Determine an, the number of words of length n on the alphabet {a, b, c} which do not contain the substring ab. For instance, a3 = 21 since there are 21 such words with 3 letters, namely: aaa aac aca acb acc baa bac bba b…

Let α, β be roots of the equation x^2− 3x − 1 = 0. For each nonnegative integer n, let y_n = α^n + β^n . Show that gcd(y_n, y_(n+1)) = 1 for each nonnegative integer n.

Determine an, the number of words of length n on the alphabet {a, b, c} which do not contain the substring ab. For instance, a3 = 21 since there are 21 such words with 3 letters, namely: aaa aac aca acb acc baa bac bba b…

Q1. Let {1,2,3,4,6,9} with the partial order of divisibility. Draw its Hasse Diagram Q2. [{1,2,3,4,5}, s], Draw its Hasse Diagram

[{1,2,3,4,5}, <=], Draw its Hasse Diagram.

Find (showing your reasoning) a formula for the term tn of the sequence defined by t1 = 4; tn = 2tn−1, n ≥ 2.

Consider the following two arguments: A. If Fred wanted me fired then he would go to the boss. Fred is going to the boss. Therefore Fred wants to get me fired. B. Max is a cat or Max is a mammal. Max is a cat. Therefore Max is not a…

Question #260486 1. Represent the following sets in tabular or listing form: (i) A={x:x^2 - 3x + 2 = 0} (ii) B= { x:x is an integer and 1 < x < 7}

∅ ⊆ A (true or false) 2.n(∅) = 0 (true or false)

Consider the sets A and B, where A = {3, |B|} and B = {1, |A|, |B|}. What are the sets?

For this arguments, explain which rules of inference are used for each step. Same body in this class enjoys whole watching. Every person who enjoys whole watching cares about ocean pollution. Therefore, there is a person in this class…

in a class of final year student in a school, 350 offer chemistry and 200 offer physics. if 90 student offer neither chemistry nor physics; i: how many students are offering both subject. ii: how many students are offering only …

Encrypt the message ATTACK using the RSA cryptosystem with n = 43 · 59 and e = 13, translating each letter into integers and grouping together pairs of integers, as done in example 11 in the textbook and in the classnotes

Using Binomial Theorem, give the closed form expression for: \textstyle\sum_{k=0}^n∑ k=0 n {n \choose k}( k n )3n · 2k

Answer the following: (a) You have a standard 52 card deck. If you draw two cards at random, what is the probability that they are both hearts? (b) You reshuffle your cards into the deck, but unfortunately now a mischievous dog (a g…

How many solutions in non-negative integers are there to the equation x1 + x2+ x3 + x4 = 19

Consider all strings of length 12, consisting of all uppercase letters. Letters may be repeated. Please do not simplify your answers. (a) How many such strings are there? (b) How many such strings contain the word ”SCOOBY”? (c) How…

State and prove Pascal’s identity using the formula for {n \choose k}( k n )

The English alphabet contains 21 consonants and 5 vowels. How many strings of five lowercase letters can be formed using the following constraints? Give two answers for each of the following - one where repetition is allowed in the st…

Find the inverse of 55 modulo 7 by using extended Euclidean Algorithm step by step solution

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

Let p and q be the propositions p : It is below freezing. (including negations). q : It is snowing Write these propositions using p and q and logical connectives a) It is below freezing and snowing. b) It is below f…

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. 2. De…

1.Determine for which integer values of n, 3n^3+2≤n^4 and prove your claim by mathematical induction. 2.

~(pVq)→r

(𝑝 → 𝑞) ↔ (𝑞 ∨ ~𝑝)

at one point in the illinois state lottery lotto game, a person was required to choose six numbers (in any order) among 44 numbers. in how many ways can this be done?

. 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 number. b) student identification numbe…

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…

2. Equal sets are always equivalent but equivalent sets may not be equal. Justify

Function f:X -> Y is called one-to-one if and only if \forall a,b\in X:f(a)=f(b)\to a =b∀a,b∈X:f(a)=f(b)→a=b a) Y is the set of mobile phones. Since two different students cannot have identical mobile phone number, the…

. 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 number. b) student identification numb…

Find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which it is false

In a company, there are 50 employees and some committees. If each employee belongs to 6 committees and each committee consists of 10 people, how many committees are there?

Find the number of positive integers less than 601 that are not divisible by 4 or 5 or 6

Prove by mathematical induction. 1 + 5 + 9 +...+ (4n-3) = (2n+1)(n-1) for n ≥ 2

Greatest and smallest elements of poset 2,4,5,10,12,20,25

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(♦…

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Question ID: mtid-5-stid-8-sqid-3313-qpid-2012