Solved: Solve the following. (35 pts) 1. How many 3-digit number can be formed from digits 1 – 5 if: a. If repetition is allowed? (3 pts) b. If repetition is not allowed? (3 pts) 2. How many strings of length 4 can be formed using the letters COMPUTER if repetitions are not allowed? (3 pts) 3. Out of 17 regions in the country, ten will be chosen to be included in a survey. How many ways of selecting 10 out of 17 regions? (3 pts) 4. Expand the (2𝑥 − 𝑦)3 using binomial coefficient. (14 pts) 5. Find for the coefficient of the following: (3 pts each) a. x4y3 after the expansion of (x – 2y )7 b. x6y6 after the expansion of (2x + y )12 6. Find the 9th term of the expression (2x + y )12 (3 pts)

Give an example of two uncountable sets A and B with a nonempty intersection, such that A−B is (a) Finite (b) Countably infinite (c) Uncountably infinite

Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f: A→B is both one-to-one and onto.] (a) f: Z × Z→Z, f(n, m) = n2 + m2 (b) f: R→R, f(x) = x3 − 3 …

Let X=Y = R and Z = set of all integers. Let f:Y-Z defined by fix) = [x], and g:X-Y be defined by g(x) = (5x-3)/2 Find: 1. f(-5/2) 2. f(n) 3. g(1) 4. fo g (x)

Let S = {Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Royal Tigers, Dundee United, Lyon} be a universal set, A = {Southend, Liverpool, Maroka Swallows, Royal Tigers}, and B = {Barn…

2. Use set builder notation to give a description of each of these sets. a) {0, 3, 6, 9, 12} b) {−3, −2, −1, 0, 1, 2, 3} c) {m, n, o, p}

out of 300 students taking discrete mathematics 60 take coffee, 27 take cocoa ,36 take tea ,17 take tea only ,47 take chocolate only ,7 take chocolate and cocoa ,3take chocolate, tea,and cocoa ,20 take cocoa only ,2 take tea,coffee an…

In the given picture are three men: Neil Armstrong, Michael Collins and Buzz Aldrin. They were on the Apollo 11 that set the first man on the moon in 1969. Neil Armstrong was the first man walking on the moon. Which is an example of…

Determine the validity of the following argument. “To pass the Discrete Mathematics, it is necessary to pass both the course work and the final examination. Either John will have to work hard in Discrete Mathematics or he will fai…

SOLUTION a) ∀x(x2 ( x) solution x=2,y=1,x\not= \frac{1}1x=2,y=1,x  = 1 1 b) ∀x(x2 ( 2) solution x=1,y=20, \\20^2-1>100x=1,y=20, 20 2 −1>100 c) ∀x(|x| > 0 solution x=8,y=4,\\8^2=4^3x=8,y=4…

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers. a) ∀x(x2 ( x) b) ∀x(x2 ( 2) c) ∀x(|x| > 0

What is truth table of logical operation (AND)and operation (OR) for five variables (five input)?

Construct a truth table for each of these compound statements.( p \leftrightarrow(p↔ q) \to(\lnot p \leftrightarrow q )→(¬p↔q)

Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f : A → B is both one-to-one and onto.] f : R × R → R, f(n, m) = 2m − n

Conventional way of writing Classify the following either mathematical expression (ME) or mathematical sentence (MS) 1. 5>8 2. a+b 3. x+y= a+b 4. t/100 5. 3. 141627980643891545746549

Let W(x, y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence.

Let P(x) be the statement ”x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in English.

1. Let p and q be the propositions p: It snowed. q: Eve goes skiing. Express each of these propositions using p and q and logical connectives. (a) It snowed but Eve does not go skiing. (b) Whenever Eve goes skiing, it snowed. (c…

3.State the converse, contrapositive and inverse of the conditional statement: When it’s hot out, it is necessary that I eat ice cream. 4. Steve, Bill and Larry go to a bar. The bartender asks: “Does everyone want beer?” Steve says…

6. Show that (p∧q) → r and (p → r)∧(q → r) are not logically equivalent, without using truth tables. 7. Express each of these statements into logical expressions using predicates, quantifiers, and logical connectives. Let the domai…

8. In the previous problem, put a negation in front of the logical expression for “Someone in your class is perfect”, then move the negation until negation only appears directly in front of S(x) or P(x), by applying DeMorgan’s Laws.…

2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 5).(5, 1), (3, 5). (5, 3), (1, 3), (3, 1)) Determine if the following is an equivalence relation on X = (1, 2, 3, 4, 5). If the following are equivalence relation, then enumerate its e…

List the members of the equivalence relation on (1, 2, 3, 4) defined by the following partition. Find the equivalence classes [1], [2], [3] and [4] 1. {(1, 2), (3, 4)) 2. {{1, 2, 3), (4)) 3. {{1}, {2}, {3}, {4}}

Let R₁ and R₂ be equivalence relation on X. Show that R₁ R₂ is an equivalence relation on X.

ASAP Determine all the winning coalitions and find the Banzhaf power distribution [16:5,5,11,6,3] SHOW ALL WORK

Given: [10.5: 5,5,6,3] 1) Determine all the sequential coalitions and find the Shapley-shubik power distribution. Show all work. 2) Does an electoral college system seem like a fair method based on the your results for shapley s…

R3 = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4), (3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)} Determine whether the relation R3 is reflexive, symmetric, anti-symmetric and transitive. Determine whether the relation …

1. Determine whether the following is a set or not a set. a. The list of course offering of UPHR-Molino Campus. -SET b. The elected barangay officials of Bacoor City. -SET c. The collection of intelligent students of College Depa…

How many rows appear in a truth table for each of these compound propositions? a) p → ¬p b) (p ∨ ¬r) ∧ (q ∨ ¬s) c) q ∨ p ∨ ¬s ∨ ¬r ∨ ¬t ∨ u

3. Let p and q be the propositions "The election is decided" and "The votes have been counted" respectively. Express each of these compound propositions as an English sentence. a) -p b) p Vqc)-p/qd)q-pe)-q→pf) pq g…

5. Construct a truth table for each of these compound propositions. a) p→ (-q V r) b) -p → (q→r) c) (pq) v (pr) d) (p→q)^(p-1) e) (pq) V (q→1) f) (p →→q) → (q→1)

6. Express these system specifications using the propositions p "The message is scanned for viruses" and q "The message was sent from an unknown system" together with logical connectives (including negations). a) &…

(4). Write the converse, inverse, and contrapositive of the statement “If 5 is an odd number, then it is a prime number.” (5). Draw a truth table and determine for what truth values of p and q the proposition ∼ q ∨ p is false. …

III. Determine the truth value of each of these statements if the domain consists of all integers. State your reason. 1. ∀𝑥, (𝑥 2 > 𝑥) 2. ∃𝑦, (𝑦 < 𝑦 2 − 1) 3. ∀𝑦, (𝑦 2 ≠ 𝑦) 4. ∃𝑥, 𝑦, (4𝑥 > 5𝑦) where 𝑥 < 𝑦 5. ∀𝑥, 𝑦, (𝑥𝑦 > 0) where 𝑥 = y

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.

Show that if n | m, where n and m are integers greater than 1, and if a≡b (mod m), where a and b are integers, then a≡b (mod n).

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

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Question ID: mtid-5-stid-8-sqid-2988-qpid-1687