Solved: (2) How many different license plates can be made consisting of 10 characters if the first 5 characters consist of uppercase English letters, while the final 5 characters are digits?

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.

Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3, 3)} c) {(1, 1), (1, 2), (1, 3), (2, 2), (…

Draw the directed graphs representing each of the relations on {1, 2, 3, 4} a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)} c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (…

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) …

What are the values of the following expressions? ∑_(i=3)^6▒ (i^2+6) ∏_(j=1)^5▒ (j/2) 4! 0! (6¦2) C(6,2)

Show that if n is a positive integer with n ≥ 3, then C(n, n − 2) = ((3n − 1)C(n, 3))/4

A gumball machine contains 300 grape flavored balls, 400 cherry flavored balls, and 500 lemon flavored balls. What is the probability of getting 1 grape ball, 1 cherry ball, and 1 lemon ball if each ball was removed and then replaced …

Let A = {2, 4} and B = {2, 4, 6} and define a relation R for A to B as follows: Question: Which ordered pairs belongs to relation R?

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime implicants, and essential prime implicants of F(x, y, z).

1. Classify and compare the different types of antenna using ah venn diagram. Discuss each type in each petal of the diagram.

1a) Use set-builder notation to prove that A-(B ∩C)=(A-B)∪(A-C) b) Let E={ 1,23,4,5,6,7,8,9,10} and ordering of element in increasing order that is a= i; what bit strings represent the subsets of integers not exceeding 5 in E …

Describe any three inference rules

Write short note on Caesar cipher with example

Show that every simple finite graph has two vertices of the same degree

Define spanning trees in a weighted graph, with an example

Show that a tree with n vertices has exactly n-1 edges

Let the operations * and ⊕ be defined on the set of integers. Find each of the following are commutives, associatives, left-distributive, and right-distributive. 1. a*b=ab; a⊕b=-a-b 2. a*b=a+2b; a⊕b=a-2b 3. a*b=2ab; a⊕b=2a+2…

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation defined by aOb = max{ a,b}

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

Write each of these statements in the form “if p, then q” in English. a) It snows whenever the wind blows from the northeast. b) The apple trees will bloom if it stays warm for a week. c) That the Pistons win the championship im…

A house which is valued at 2,600, 000 appreciated at rate of 12% per year I. What will be its value after two and half years? ii. After how long will its value be 4200,000

Consider the following series 56, 28, 14.. I. Find 17th term. ii. Find the sum of the series if it continues indefinitely iii. Find 20th term.

Show that (p → r) ∧ (q → r) and (p ∨ q) → r are logically equivalent.

Show that each of these conditional statements is a tautology by using truth tables. a) (p ∧ q) → p b) p → (p ∨ q) c) ￢p → (p → q) d) (p ∧ q) → (p → q) e) ￢(p → q) → p …

What is the truth value of ∀xP(x), where P(x) is the statement “x2 < 10” and the domain consists of the positive integers not exceeding 4?

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.] a) It is necessary to wash the boss’s car to get promoted. …

Suppose you want to encode messages containing only the following characters with their given respective frequencies: B: 55 D: 15 E: 80 G: 5 U: 45 (a) What is the minimum length bit string required to encode each character with a dis…

Let us take a connected multi graph G with 2k vertices of odd degree and go in reverse direction for the proof. Initially we start pairing the vertices of odd degree and go on adding an extra edge joining the vertices in each pair, i…

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 …

Consider the nonhomogeneous linear recurrence relation an = 3an−1 + 2^n. Show that a^n = – 2^(n+1) is a solution of this recurrence relation.

Determine values of the constants A and B such that an = An + B is a solution of recurrence relation an = 2an−1 + n + 5. Hence, find the solution of this recurrence relation with a0 = 4.

Pick a number. Add 4 to the numbers, and multiply the sum by 2 subtract 5 and decrease this difference by twice the original numbers (hint let n represent the original numbers and 2n be twice

List five situation from daily life in which graphs arise naturally

j=0 ∑ 8 ( j 8 ) (j+1)(j+2) 1 =1⋅ (0+1)(0+2) 1 +8\cdot\dfrac{1}{(1+1)(1+2)}+28\cdot\dfrac{1}{(2+1)(2+2)}+8⋅ (1+1)(1+2) 1 +28⋅ (2+1)(2+2) 1 +56\cdot\dfrac{1}{(3+1)(3+2)}+70\cdo…

∑j=08(j8)(j+1)(j+2)

Consider series 56,28, 14.. I. Find 17th term Ii. Find the sum of series if it continues indefinitely. Iii.find 20th term Ii. Find the sum of series if it continues indefinitely. Iii.find 20th term

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

Let f : A → B be a function. 1. Show that for the identity function iA on A we have f ◦ iA = f. 2. Show that for the identity function iB on B we have iB ◦ f = f.

Show that a tree with n vertices has exactly n-1 edges

A. Try these! 1. Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}

1. Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}.

Let f(x) = x – 3 , g(x) = 2x + 1 and h(x) = x2 – 5. Find the following 1. f ₒ g 2. f ₒ h 3. h ₒ g 4. g ₒ f 5. g ₒ h

B. Exercises: Let f(x) = x – 3 , g(x) = 2x + 1 and h(x) = x2 – 5. Find the following 1. f ₒ g 2. f ₒ h 3. h ₒ g 4. g ₒ f 5. g ₒ h

Determine the domain of each of the following functions: 1. f(x) = x + 10 6. A(x) = x2 -2 2. F(x) = 2 3 𝑥 + 5 7. H(x) = √𝑥 − 2 3. g(x) = 5 – 3x 8. K(x) = √𝑥 2 − 2 4. g(x) = 1 (𝑥+5)(𝑥−1) 9. C(x) = 2x3 + 4x2 - 2x + 1 5. b(x) = 𝑥−1 𝑥 2+5…

Let A, B and C denotes the subset of a set, S and let 𝐶̅denotes the complement of C in set S. If (A ∩ C) = (B ∩ C) and (A ∩ 𝐶̅) = (B ∩ 𝐶̅), then prove that (A = B).

Given A = {x | x ∈ Z- ∧ 3x > -10}. Determine elements of A.

Let the universe of discourse be the set of all integers. Let p; q; r; s, and t be as follows: p(x):x>0,q(x):xiseven,r(x):xisaperfectsquare,s(x):xis(exactly)divisibleby4, t(x):x is (exactly) divisible by 5. (8 marks) Write the follow…

Construct a truth table for each of these compound propositions. (i) (p → q) ↔ (¬q → ¬p) (16 marks) ii) p ⊕ (p ∨ q) (8 marks) (i) Determine by using truth tables if (p ∧ q) → p is a tautology, contradiction or a contingency. Give…

How many ways are there to select 12 countries in the United Nations to serve on a council if 3 is selected from a block of 53, 1 are selected from a block of 62 and 8 are selected from the remaining 74 countries?

(2) How many different license plates can be made consisting of 10 characters if the first 5 characters consist of uppercase English letters, while the final 5 characters are digits?

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Question ID: mtid-5-stid-8-sqid-3592-qpid-2291