Solved: a) Let p, q, and r be the propositions p : You study hard. q : You are on merit r : You do not get degree from Iqra University. Express each of these propositions as an English sentence. i) ¬q ↔ r ii) q →¬ r iii) p∨q ∨¬r iv) (p →¬r)∨(q →¬r) v) (p∧q)∨(¬q ∧¬r)

1. Expand the following using the Binomial Theorem a. (x-3y)^3 b. (2a+b^-3)^4 c. (a+b+c)² 2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

P = I pass this class G = I go to class every day. H = I do all the homework exercises. Translate the following sentences into propositional logic. (1) Students will not pass this class unless they go to class every day and do all…

Solve the following recurrence relations for the initial conditions given. The population of Utopia increases 5 percent per year. In 2000 the population was 10,000. What was the population in 1970?

Write the negation of the following statement: ∀𝑥∃𝑦(𝑥 + 𝑦 = 2 ∧ 2𝑥 − 𝑦 = 1

Suppose that the domain of the predicateP(x) consists of −2, −1, 0, 1, and2. Write out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

illustrate the venn diagram for the sets A,B,and C in the U={1,2,3,4,5,6,7,8} a={1,2,3,4,5,6,7}, b={1,5,6,7}, c={1,2,3,6}

Q1: Let A, B, C be three sets. Prove that (𝐴 − 𝐵) − 𝐶 = (𝐴 − 𝐶) − (𝐵 − 𝐶) using following proof methods: a) Membership table b) Set builder notation and propositional logic Q2 : Find generalized union and intersection as de…

Draw Hasse diagram repressing the partial order on {(a,b) : a\b } on {1,2,3,4,6,8,12}

I. Create a truth table for all of the 11 Logical Equivalence. II. Find if the following are logically equivalent 1.(~p v q) ^ (~q) <=> ~(p v q) 2.(p ^ ~q) v (~p v q) <=> T

29. Prove or disprove that if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

Let A and B be subsets of a universal set U. Show that A ⊆ B if and only if B ⊆ A.

Use rules of inference to show that the hypotheses ”If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,”, ”If the sailing race is held, then the trophy will be awa…

6. (a) Draw the logic circuit for the following expression: AB + A(B+C) (b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A computer randomly chooses a binary string of length 6. What is the probability that exactly two characters in the string turn out to be 1s?

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A box contains 6 red, 8 green,10 black 11 yellow and 12 white balls. What is the minimum number of balls we have select from box to guarantee that 9 balls are of the same colours.

Prove that if 10 points are placed in a 3cm by 3cm squre board than two points must be at least √2 cm apart.

6. (a) Draw the logic circuit for the following expression: AB + A(B+C) (b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A weighted voting system for voters A, B, C, D, and E is given by {35: 29, 11, 8, 4, 2}. The weight of voter A is 29, the weight of voter B is 11, the weight of voter C is 8, the weight of voter D is 4, and the weight of voter E is 2.…

A toy shop has 15 airplanes, 15 buses, 17 trains, and 20 bikes in the stock. a. How many ways are there for a person to take 15 toys home if all the airplanes are identical, all the buses are identical, all the trains are identical an…

Let P: It is below freezing and Q: It is snowing. What the symbolic form of the "It is not snowing, it is below freezing".

Prove that in a graph G, the sum of the degrees of the vertices is equal to twice the number of edges. Consequently, the number of vertices with odd degree is even

⋃ 𝐴𝑖 ∝ 1 and ⋂ 𝐴𝑖 ∞ 1 a) 𝐴𝑖 = {𝑖, 𝑖 + 1,𝑖 + 2, … } b) 𝐴𝑖 = {0,𝑖} c) 𝐴𝑖 = {−𝑖, − 𝑖 + 1, … , −1, 0, 1, … , 𝑖 − 1, 𝑖} d) 𝐴𝑖 = {−𝑖, 𝑖}

Check whether the following graphs have an Eulerian and/or Hamiltonian circuit.

Let P (x) denote the statement “x ≤ 4.” What is the truth value of p(6) ?

Find the bitwise “and” 0110 1101 , 1001 0011

Mathematical logic

What and Define Conjunction

What and Define Disjunction of OR and Exclusive

What and Define Negation

What and Define conditional and biconditional operator

What and Define biconditional

What and Define Tautology and contradiction

What and Define Equivalent formula or laws of algebra of propositions

What and define Rules of inferences

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.

a) Let p, q, and r be the propositions p : You study hard. q : You are on merit r : You do not get degree from Iqra University. Express each of these propositions as an English sentence. i) ¬q ↔ r ii) q →¬ r iii) p∨q ∨¬r iv) (p →¬r)∨(q →¬r) v) (p∧q)∨(¬q ∧¬r)

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Question ID: mtid-5-stid-8-sqid-3511-qpid-2210