Solved: Write each statement in symbolic form, using the given symbols. Let p: I will study. Let q: I will pass the test. Let r: I am foolish. 1. I will study or I am foolish. 2. I will study or I will not pass the test. 3. I will study and I will pass the test. Page 1 MATHEMATICS IN THE MODERN WORLD 4. I will past the test or I am foolish. 5. I am not foolish and I will pass the test. 6. I will not study or I am foolish. 7. I will study or I will not study. 8. I will study and I will pass the test, or I am foolish. 9. 9. It is not true that I will study or I am foolish. 10. It is not the case that I will not study or I am not foolish.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? How many must be selected to guarantee that at least 3 hearts selected?

During a month with 30 days, a base ball team plays at least one game a day but no more that 45 games. Show that there must be a period of some number of consecutive days during which the team must plays exactly 14 games.

Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail “. There are 35 students pass. Show that there are at least two students pass in the list exactly nine students apart. (for exam…

Suppose that you meet three people Aiman, Borhan, and Camelia. Can you determine what Aiman, Borhan, and Camelia are if Aiman says "All of us are knaves" and Borhan says "Exactly one of us is a knave."?

The recursive algorithm given below can be used to compute gcd(a, b) where a and b are non-negative integer, not both zero. procedure gcd(a, b) if a > b then gcd(a, b) := gcd(b, a) else if a = 0 then gcd(a, b) := b…

7. (a) Use the rules of Boolean algebra provided in (5) to simplify the following logic gate circuit: (b)Draw the simplified logic gate circuit in (a)

Determine whether each following statements about Fibonacci numbers is true or false A. 2Fn >F n+1 for n≥3 B. 2F n+4 = f n+3 for n≥3

Consider a recurrence relation an = -3an-1 + n for n = 1,2,3,4,… with initial conditions a1 = 3. Calculate a3.

Consider a recurrence relation an = an-1 - 3an-2 for n = 1,2,3,4,… with initial conditions a1 = 3 and a2 = 5. Calculate a5.

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail “. There are 35 students pass. Show that there are at least two students pass in the list exactly nine students apart. (for exam…

State the Pigeonhole Principle. A chess player wants to prepare for a championship match by playing some practice games in 77 days. She wants to play at least one game a day but no more than 132 games altogether. Prove that there is…

b) Given p="Front door is open q="Back door is open" r="Light is on" s="The lift doors are open" t="The lift is moving", i) write the follow…

When considering the set of all the natural numbers (ℕ), show whether the mathematical operations of addition, subtraction, multiplication and division are: (a) Associative (b) Commutative.

Given that 𝐺 = {𝑎 ∈ ℝ|𝑎 ≠ −1} and 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏, show that (𝐺, ∗) is indeed a group.

You are required to prepare a 15-minute presentation aimed towards new students engaged under a technology company’s graduate scheme. Your presentation will include evidence of: 1. An explanation that adequately explains why group t…

Suppose a recurrence relation an=4an−1−4an−2 where a1=14 and a2=40 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where …

a n −4a n−1 +4a n−2 =n 2 \displaystyle\sum_{n=2}^{\infin}a_nx^n-4\displaystyle\sum_{n=2}^{\infin}a_{n-1}x^{n}+4\displaystyle\sum_{n=2}^{\infin}a_{n-2}x^{n}=\displaystyle…

Use generating functions to solve the recurrence relation an = 4an−1 − 4an−2 +n2 , where a0 = 2, a1 = 5.

Answer these questions for the poset ({3, 5, 9, 15, 24, 45}). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of { 5, 9}. f ) Find…

Find the power set of each of these sets, where a and b are distinct elements. a) {a} b) {a, b} c) {∅, {∅}}

Show that if x is a real number, then ⌈x⌉ − ⌊x⌋ = 1 if x is not an integer and ⌈x⌉ − ⌊x⌋ = 0 if x is an integer.

Show that if x is a real number, then x − 1 < ⌊x⌋ ≤ x ≤ ⌈x⌉ < x + 1.

Use a predicates and quantifiers to express this statement. There is a man who has visited some park in every state of malaysia

solve the recurrence relation Pn = Pn-1 + n with initial condition P1 = 2 using interation.

Problem 1. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 20 salespeople given the information below. Shift Average number of customers…

how to solve power set? {12,34,{45}, 56}

Suppose that the universal set is U = {1, 2, 3, 4,5, 6, 7, 8, 9, 10}. Express each of these sets with bit strings where the ith bit in the string is 1 if i is in the set and 0 otherwise.

Counting subsets of Finite sets: A student can choose a computer project from one of three lists. The three lists contain 23, 15 and 19 possible projects, respectfully. No project is on more than one list. How many possible …

An investor is considering a $25,000 investment in a start-up company. She estimates that she has probability 0.05 of a $15,000 loss, probability 0.15 of a $20,000 loss, probability 0.35 of a $35,000 profit, and probability 0.45…

For all integers n and m, if n − m is even then n^3 − m^3 is even.

Prove the statement by contraposition, if a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.

Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each n…

Describe at least one way to generate all the partitions of a positive integer n.

Proof that an undirected graph has an even number of vertices of odd degree.

Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in …

Suppose a recurrence relation an=2an−1−an−2 where a1=7 and a2=10 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and y are roots of the ch…

Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each n…

a. How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of four aces (A) are chosen?

Which of the following pairs of statements are logically equivalent? I. Statement 1: It is not true that both you have pneumonia and I got flu. Statement 2: Either you have no pneumonia or I have no flu. II. Statem…

Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in …

a) Show or verify that 3 is a primitive root of the prime p=7. b) Show that 2 is not a primitive root of 7. c) Show the steps of the Diffie-Hellman key agreement protocol between Alice and Bob, assuming they use the prime 7 and its p…

5 a) Use merge sort to sort 6, 5, 17, 15, 16, 20, 18, 7 into increasing order. Show all the steps in the algorithm (draw the trees as shown in class). b) What is the number of comparisons used in the merge algorithm shown in class t…

8 A bag of chocolates has 20 milk chocolates, 20 dark chocolates and 20 white chocolates. a) What is the minimum number of chocolates you must select at random from the bag to guarantee that you will get at least 5 of the same kind …

Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each n…

1. Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) Fo…

2. Describe at least one way to generate all the partitions of a positive integer n. (You can get idea from Exercise 49 in Section 5.3.)

3. Proof that an undirected graph has an even number of vertices of odd degree.

4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists. i) ii)

4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument 5. Suppose that G is a connected multigraph with 2k …

State the Pigeonhole Principle. A chess player wants to prepare for a championship match by playing some practice games in 77 days. She wants to play at least one game a day but no more than 132 games altogether. Prove that there is…

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail “. There are 35 students pass. Show that there are at least two students pass in the list exactly nine students apart. (for exam…

Construct a truth table for (~p ˄ r) ˅ (q ˄ ~r).

A survey of 119 persons was conducted at TTC, and it was found that 89 persons carried a cell phone, 10 person carried a tablet computer, and 8 carried both cellphone and tablet. how many people carried a cellphone or a tablet?…

Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive. (x, y) ∈ R if x ≥ y, where R is the set of positive integers.

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with 0 < a < b < c < d < 8.

Suppose a recurrence relation an=2an−1−an−2 where a1=7 and a2=10 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and …

Draw the directed graph that represents the relation {(a,a), (a, b), (b, c), (c, d), (a, d), (b, d), (d, b)}.

Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0,1),(1, 1), (1, 2), (2, 0), (2, 2) and (3, 0). Find the (i) reflexive closure of R, (ii) symmetric closure of R

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with 0 < a < b < c < d < 8.

2. Use generating functions to solve the recurrence relation an = 4an−1 − 4an−2 + n2, where a0 = 2, a1 = 5.

Use a direct proof to show that every odd integer is the difference of two squares.

Use a proof by contraposition to show that if 𝑛𝑛2 + 1 is even, then 𝑛𝑛 is odd.

Let 𝑃𝑃(𝑛𝑛) be the proposition that 1(1!) + 2(2!) + 3(3!) + ⋯+ 𝑛𝑛(𝑛𝑛!) = (𝑛𝑛+ 1)! −1. Prove by induction that 𝑃𝑃(𝑛𝑛) is true for all 𝑛𝑛≥1.

In how many different ways can you put20 balls of the same colour into two numbered box

Determine whether each of the following is true or false. a. 0 ∈ ø b. ø ∈ {0} c. {ø} ⊆ {0} d. {ø} ⊆ {ø} e. ø ∈ {0, ø}

5. Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an e…

3. Proof that an undirected graph has an even number of vertices of odd degree.

The negation of the statement " 52 is divisible by 4 and 7" is

What is nested Quantifier? Is order important for nested quantifier? Explain your answer with appropriate example.

Use rules of inference to show that the hypotheses “If the weather is not too hot or not too cold, then the game will be held and a prize-giving ceremony will occur,” “If the game is held then the VC will give a speech,” “The VC di…

Use generating functions to solve the recurrence relation an = 7an−1 − 16an−2 + 12an−3 + n4n , where a0 = −2, a1 = 0, a2 = 5.

What is the units digit in the expansion of 2 to the power of 727 ? 8 4 2 6

A magic square is an arrangement of n2 numbers into n rows and n columns using distinct numbers from 1 up to n2 such that the sum in any row, any column or any of the two diagonals is fixed. Consider the 4 by 4 magic square below: a…

Oliveira, Anibal, Julia and Andres are planning to rob a bank. To hide their true identities, they came up with the idea of using the names of cities as aliases, namely, Manila, Tokyo, Rio and Berlin. Berlin is twice as old as Julia. …

{x | x is a real number such that x2 = 1}

draw the hasse diagram for the poset({1,3,6,9,12}) hence determine whether it is a lattice

What is nested Quantifier? Is order important for nested quantifier? Explain your answer with appropriate example.

In how many ways a relation can be represented? State two different examples to represent each of them.

Suppose a recurrence relation an=7an−1−12an−2 where a1=16 and a2=52 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and y ar…

Show that the relation R = ∅ on a nonempty set S is sym- metric and transitive, but not reflexive.

Show that the relation R = ∅ on the empty set S = ∅ is reflexive, symmetric, and transitive.

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