Solved: Suppose U = {1, 2, 3, a, b, c} is a universal set with the subset A = {a, c, 2, 3}. Let R = { (a, a), (a, c), (3, c), (3, a), (2, 3), (2, a), (2, c), (c, 2) } be a relation on A. Answer questions 11 & 12 by using the given sets A, U and the relation R. Question 11 Which one of the following statements regarding relation R is true? 1. R is a weak total order. 2. R is a strict total order. 3. R is a weak partial order. 4. R is not an equivalence relation. Question 12 Which ordered pairs should be removed from relation R in order for the changed relation R1 (say) to be a strict partial order? 1. only (2, c) 2. only (a, a) 3. (a, a) & (c, 2) 4. (a, a) & (2, c)

(i). In MA161 class, the lecturer gave the following proposition. Let A (x, y) be “ y is greater than or equal to x ”. The domain for the proposition is the set of nonnegative integers. The students were tasked to determine what are t…

Draw a Venn diagram that represents V, the set of vowels in the English alphabet, where set of vowel is {a,e,i,o,u}

Let A = {0, 2, 4, 6, 8}, B = {0, 1, 2, 3, 4}, and C = {0, 3, 6, 9}. What are A ∪ B ∪ C and A ∩ B ∩ C?

How many ways to arrange the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in two(2) rows of six (6) columns so that (1) the integers increase in value as each row is read, from left to right and (2) in any column the smaller integ…

I. Use quantifiers to express the following statements. 1. Every student enrolls in a math class. 2. There is a student in a class who drops a math course. 3. Every student in this class wears an ID. 4. At least one student in thi…

Let P(x) be the statement “x = x²." If the domain con- sists of the integers, what are these truth values? c) P(2) f) VxP(x) a) P(0) b) P(1) d) P(-1) e) 3x P(x)

Draw these graphs. a) K7 b) K1,8 c) K4,4 d) C7 e) W7 f ) Q4

Can a simple graph exist with 15 vertices each of degree five?

Write these propositions in symbols using p, q, and r and logical connectives. You get a 95 in math, but you do not do every assignment in the class

A. Use quantifiers to express the following statements. 1. Every student enrolls in a math class. 2. There is a student in a class who drops a math course. 3. Every student in this class wears an ID. 4. At least one …

Let A = {2, 4} and B = {6, 8, 10} and define relations R and S from A to B as follows: for all (x,y) EA x B. XRyoxly for all (x,y) EA x B x Syy-4 = x State explicitly which ordered pairs are in A x B. R, S, RUS and Ros. And draw the C…

R1 = {(4,5)} R2 = {(1,5), (1,6), (1.7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)} Evaluate R1 ◦ R2 and R2 ◦ R1

Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers not exceeding 10.

If (S,*) is a.semigroupand x € s show that (S,∆) is a semigroup if a∆b =a*x*b

Verify the validity of the argument: All lions are fierce. Some lions do not drink coffee. Hence some fierce creatures do not drink coffee.” (Lewis Carrol)

LetA={0,2,4,6,8},B={0,1,2,3,4},andC={0,3,6,9}.Find the following: A∪B∪C A∩B∩C (A∪B∪C)𝐶 4.(A∩B∩C)𝐶 5.(A∪B)∩C)𝐶

mathematical notations for The set of all even numbers.

Use set builder notation to give a description of each of these sets. a. {0, 2, 4, 6, 8,10,12,14,16} b. {−3,−2,−1, 0, 1, 2, 3}

how many ways to arrange the integers 1,2,3,4,5,6,7,8,9,10,11,12 in two rows og six columns so that 1 the integers increase in value as each row is read, from left to right and 2 in any column the smaller integer is on top.

Find out if the following sets are Countable, Uncountable, Finite or if it cannot be determined. Give the reasoning behind your answer for each. (a) Subset of a countable set (b) integers divisible by 5 but not by 7 (c) (3, 5) (…

(a) Find the solution to an = an-1 + 2n + 3 with the initial conditions a0= 4. (b) Consider the recurrence an = an-1 + 2an-2 + 2n - 9 show that this recurrence is solved by: i. an = 2 - n ii. an = 2 - n + b * 2n for any real b.

Given two sets A and B, for each of the following statements, what can you conclude about the sets? For example: consider the statement A−B=∅, this could be possible if - Scenario - 1: if A=B then A−B=∅ Scenario - 2: Since A−B=A−(A…

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 …

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Question ID: mtid-5-stid-8-sqid-2963-qpid-1662