Solved: A. COUNTING METHODS 1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not allowed? 2. There are 10 multiple choice questions in an examination. Each of the questions have four choices. In how many ways can an examinee give possible answers? B. BINOMIAL COEFFICIENTS Expand (2𝑥 + 4𝑎) 4 using the binomial theorem. C. PIGEONHOLE PRINCIPLE (5 pts) . Explain briefly. Do you agree that there are 3 persons who have the same first and last name? Why and why not?

Find a div b and a mod b when: (a) a = 30303, b = 333 (b) a = −765432, b = 38271

Consider the function f(n) = 35n3+ 2n3log(n) − 2n2log(n2) which represents the complexity of some algorithm. (a) Find a tight big-O bound of the form g(n) = np for the given function f with some natural number p. What are the constan…

Find the big−O, big−Ω estimate for x7y3+x5y5+x3y7. [Hint: Big-O, big- Ω, and big-Θ notation can be extended to functions in more than one variable. For example, the statement f(x, y) is O(g(x, y)) means that there exist constants C, k…

Solve for x if (g ◦ f)(x) = 1. Here, f(x) = (xlog(x) · x2) and g(x) = log(x) + 1.

What is the big-O estimate of the function given in the pseudocode below if the size of the input is n? (a function that takes in a list of numbers as input and returns the biggest number) Justify your answer. define function(input…

(3). If p, q, and r denote the following propositions: p : 2 < 3. q : The cube of -1 is -1. r : The empty set contains one element. express the following propositions symbolically. (a) If 2 ≥ 3, then the cube of -1 is -1. …

Prove that 2^(n+1) > (n + 2) · sin(n) for all positive integers n

Consider the three couples in Chintu, Pintu and Mintu and their wives Chinky, Pinky and Minky. After the dinner party, one more couple Rinku and Rinki arrives to meet them and they all decide to dance. They all want to dance in pairs …

There are n sisters in a family. They each have one dress. They decide to exchange their dresses such that nobody wears her own dress. They try all such combinations and take pictures. All the sisters appear in each picture and the or…

Consider 10 letters word as AABBCCDETS. How many circular arrangements of the ten letters are possible such that any of the same letters do not appear consecutively

There are 6 friends Ajay, Bob, Chintu, Devi, Emily and Fatima. They want to play a game which requires three teams. (Each team must have at least one player.) In how many ways can they form such teams if Chintu does not want to be alo…

How many strings of length 4 on the alphabet {A,B,C,D} do not contain AB as a substring

How many subsets of the set {1, 2, 3, 4, 5, 6, 7, 8} do not contain two consecutive integers ?

. If I own a dog, then I own an animal. what is the inverse

Express each of these statements into logical expressions using predicates, quantifiers, and logical connectives. Let the domain consist of all people. Let S(x) be “x is in your class,” P(x) be “x is perfect.” (a) Nobody is perfect…

Let p and q be the propositions “The election is decided” and “The votes have been counted” respectively. Express ~p as an English sentence. *

Which of these sentences are propositions? What are the truth values of those that are propositions?

There are 6 friends Ajay, Bob, Chintu, Devi, Emily and Fatima. They want to play a game which requires three teams. (Each team must have at least one player.) In how many ways can they form such teams if Chintu does not want to be alo…

How many strings of length 4 on the alphabet {A,B,C,D} do not contain AB as a substring ?

There are 3 black balls, 4 blue balls and 5 red balls in a box. In how many ways can we choose 3 balls at the same time with different colors?

Find, showing all working, a recursive definition of the sequence with general term tn = 6 (n + 1)!/3n, n >= 1

The heart beat rate has to be well regulated to provide enough oxygenated blood throughout the body and so depends on feedback with the body’s oxygen demand. A simple discrete model of heart beat regulation is given by: xt+1 = kxt…

Imagine you own two shops selling cakes. You are tracking the number of cakes sold in each shop each day over a ten day period. Each cake in Shop 1 makes 3 dollars profit. Each cake in Shop 2 makes 4 dollars profit. The following a…

16. Consider the set of digits {1,3, 4,5,7,8,9} a)If the digits cannot be repeated and if the middle digit must be an odd digit, how many three-digit numbers can be formed? b)If the digits can be repeated, how many positive numbers …

At a school education fair, 50 students were interviewed about their interested programmes of study and all of them have shown interest in at least one of the three programmes in the survey. From the results of the interviews, there w…

A bank password consists of two letters of the English alphabet followed by two digits. How many different passwords are there?

Prove that for any integer n n, if n n is an odd integer, then 6n 2 +5n+1 6n2+5n+1 is an even integer.

Q: You do every exercise in the class. r: You get a 95 in MMW Write these proposistions symbols using p, q, and r, and logical connectives. 1.You get a 95 in MMW, but you do not do every exercise in the class. 2. You get a 95 on the …

Let A = {1, 2, 3, 6, 12} and R = {(a, b)/a,b A and “a divides b”}. Show that the relation R is a partial order relation and also draw its Hasse diagram

Find, showing all working, a recursive definition of the sequence with general term tn = 6 (n + 1)!/3n, n >= 1

Use mathematical induction to prove that 2n > n2 , for n > 5 .

. Which of the intervals (0, 5), (0, 5], [0, 5), [0, 5], (1, 4], [2, 3], (2, 3) contains a) 0? b) 1? c) 2? d) 3? e) 4? f ) 5?

Determine if the following is an equivalence relation on X = {1, 2, 3, 4, 5} in {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 5),(5, 1), (3, 5), (5, 3), (1, 3), (3, 1))

Build a truth table then verify if the proposition is Tautology, Contradiction, and Contingency. (p ↔ q ) Λ ( ┐p Λ q )

Imagine you own two shops selling cakes. You are tracking the number of cakes sold in each shop each day over a ten day period. Each cake in Shop 1 makes 3 dollars profit. Each cake in Shop 2 makes 4 dollars profit. The following ar…

Prove by contradiction that for any integer n if n2 is odd then n is odd.

Show following equivalence without considering the truth table. (𝑝̅ ∧( 𝑞̅∧𝑟)) ∨(𝑞 ∧𝑟) ∨(𝑝 ∧𝑟)↔𝑟

. Let p and q be the propositions p: He is rich q: He is happy Write the following propositions using p and q and logical connectives. a. If he is rich, then he is unhappy. b. He is neither rich nor happy. c. It is necessary to be …

Give the power set of the following sets. (a) /0 (b) {1} (c) {1,2} (d) {1,2,3}

~(PÃ¢ q) Ã¢ p simplify by laws

state the value of x after the statement if P(x) then x = 1 is executed , when P(x) is the statement =x>1 , " if the value of x when this statement is reached is

Find, showing all working, a formula for the n-th term tn of the sequence (tn) defined by t1 = 5; tn = −7tn−1/3, n ≥ 2.

] Carry out the following calculations and conversions by hand (showing all your working). (a) Compute the sum (1011010)2 + (11011)2 (without changing the base). (b) Compute the sum (135)8 + (2357)8 (without changing the base). (c) Co…

Which of the following are propositions? Of those which aren’t, explain why. The baby is laughing. What’s that noise? 1 + 1 = 3 Get out of here! He ran in the race but slipped on a banana peel. What a glorious day it is! Santa …

Translate the following into symbols, given; p: It is hot. q: It is wet. r: I am tired. Example: 13. It is not the case that it is either hot or wet.  ~(p∨q) 1. It is not hot. 2. It’s hot and it’s wet. 3. It is not wet and I …

If 2 ≥ 3, then the cube of -1 is -1

1. Convert each of the following to their respective Decimal, Octal, Hexadecimal and binary representation: (a) (742)8 (b) (1011)2 (c) (47)10 (d) (3EAC)16

Let a and b be two Natural Numbers, such that the greatest common divisor of a and b is 63, and the least common multiple of a and b is 44452800. If ’b’ is an odd number, what is the minimum value of ’a’ possible? [Hint: a · b = gcd(a…

Solve the following: (a) 1231001 (mod 101) (b) 17123 (mod 13)

Let {an} be a sequence that satisfies the recurrence relation an=anÃ ¢ 1+3anÃ ¢ 2, for n=2,3,.,.,., where a0=1,a1=2. Find the values of a2,a3.

Find the characteristic root of the recurrence relation an=anÃ ¢ 1+2anÃ ¢ 2.

Consider the following argument: I will get grade A in this course or I will not graduate If I do not graduate, I will join the army I got grade A. Therefore, I will not join the army Is this a valid argument?

Prove that for any propositions p,q,r the compound proposition {pÃ ¢ (qÃ ¢ r)}Ã ¢ {(pÃ ¢ q)Ã ¢ (pÃ ¢ r)} is a tautology

Consider the following argument: I will get grade A in this course or I will not graduate If I do not graduate, I will join the army I got grade A. Therefore, I will not join the army Is this a valid argument?

Build a complete truth table and Show that they are Logically equivalent ¬(¬P ∧ Q) ∧ (P ∨Q)≡ p

For integers 𝑥 and 𝑦, consider the proposition ∃𝑥∀𝑦 (𝑥 + 𝑦 = 10). Is the proposition true, false, or does it depend on the values of 𝑥 and 𝑦?

In a class of n n students (we consider all students as distinct), we want to make k k groups where each group must contain at least 1 student. Let S(n,k) S(n,k) denote the number of ways in which such groups can be formed. Which o…

How many subsets of the set {1, 2, 3, 4, 5, 6, 7, 8} do not contain two consecutive integers ?

Establish the validity of the argument with the premises p -> (q -> r) , p \/ s , t ->q , ~s and ~r -> ~t

Prove that for all integer n3, P(n+1,3)−P(n,3)=3P(n,2).

Prove that n•P(n−1,n−1)= P(n,n).

find an increase in sequence of maximal links in a decrease in sequence of a maximal legs in a sequence 22,5,7,2,23,10, 15,21,3,17

Make a Fully truth table and Show that they are logically equivalent ⌐ (⌐p ∧ q) ∧ (p ∨q) ≡ p

Build a Fully truth table and identify if tautology, Contradiction or Contingency ¬(¬(P∧Q)↔(¬P)∨(¬Q))

Suppose that the domain of the predicate P (x) consists of 1, 2, 3, 4, and 5. Write out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

Given set A = {A, B, C, D, E, F, G, H} and B = {B, D, E, J, K}. Find for the: 1. A ∪ B 2. A ∩ B 3. A x B 4. B x A

Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that “Every positive integer is the sum of the squares of four integers.

The sum of the first n positive odd integers is n2. Establish the formula for the sum of the first n positive even integers and use proof by mathematical induction to prove its correctness.

prove that AUB = AU(B\A)

Let p and q be the propositions

M is a subset of the set of natural numbers. 10 elements of the set are prime numbers, and the rest are divisible by either 2, or 3, or 5. Determine the cardinality of the set if it contains: 70 numbers that are divisible by 2; 60 num…

x = 1 What is the value of x after each of these statements a. if x+2=3 then x:=x + 1 b. if (x+1=3) OR (2x+2=3) then x:=x+1 c. if (2x +3=5) AND (3x +4= 7) then x= x + 1 d. if (x+1=2) XOR (x+2=3) then x:=x+1 e. if x < 2 then x:=x…

If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area. * r→ q ↔ ¬p

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, −3, 2, −8, −3, −13, −8, −18, ? (c) 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, ? (2) Find a pair of numbers that provides a counte…

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, Ã ¢ 3, 2, Ã ¢ 8, Ã ¢ 3, Ã ¢ 13, Ã ¢ 8, Ã ¢ 18, ? (c) 1/2, 2/3, 3/4, 4/5…

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, −3, 2, −8, −3, −13, −8, −18, ? (c) 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, ?

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a. 3x (x ² = 2) b. 3x (x ² = -1) c. Vx (x ²+2 Ã ¢ ¥ 1) d. Vx (x ² + x)

Let p and q be the propositions

Let p, q, and r be the propositions p : You have Covid. q : You miss the prelim examination. r : You pass the course. Express each of these propositions as an English sentence. a) p → q b) ¬q ↔ r c) q → ¬r d) p ∨ q ∨ r e) (p → …

What is discrete mathematocs and application

Make a Fully truth table and identify if it is a tautology, contradiction, contingency. ¬(¬(P∧Q)↔(¬P)∨(¬Q))

Use algebra of sets to prove that, [(𝐵 − 𝐴)' ∩ 𝐴] − 𝐴' = 𝐴

(4). Construct a difference table to predict the next term of each sequence. (a) 9, 4, 3, 12, 37, 84, ... (b) 10, 10, 12, 16, 22, 30, ... (7). Use Polya’s four-step problem-solving strategy and the problemsolving procedures to …

.Find these values. a) Ã ¢ 1.1Ã ¢ b) Ã ¢ 1.1Ã ¢ c) Ã ¢ Ã ¢ 0.1Ã ¢ d) Ã ¢ Ã ¢ 0.1Ã ¢ e) Ã ¢ 2.99Ã ¢ f ) Ã ¢ Ã ¢ 2.99Ã…

Let X={1,2} and Y={a,b,c}. List the elements in each set.

. Translate in two ways each of these statements into logi- cal expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. …

Prove that for all integer n>=3, P(n+1,3) - P(n,3) = 3P(n,2)

If ad-bc=0 then: a/c = b/d y = a/c = b/d So, y is a constant and is not one-to-one. Finding the inverse function: Let y = f(x) = (ax+b) / (cx+d) So y (cx+d) = ax+b So cxy + dy = ax+b So cxy - ax = b - dy So x (cy - a) = b …

Find the domain and range of the following function given by 𝑓(𝑥)=√(3𝑥−5)(𝑥+4)𝑥3−16𝑥.

Show that if 𝑎𝑑−𝑏𝑐 ≠ 0, then the function 𝑓(𝑥)=𝑎𝑥+𝑏/𝑐𝑥+𝑑 is one-to-one and find its inverse.

Prove that ( ) ( ) 1, 1 , nPnnPnn  − − = .

Show that ( ) ( ) ( ) 1, , 1 , CnkCnkCnk + = − + .

(a)How many bit strings of length 8 contain at least 6 ones? (b)How many bit strings of length 8 contain at least 3 ones and 3 zeros?

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Question ID: mtid-5-stid-8-sqid-3023-qpid-1722