Solved: PROPOSITIONAL LOGIC: Let p, q and r denotes the following statements: p: A square has four equal side q: Rectangle has 2 parallel sides r: A square is a rectangle. 1. Express each of the following into English sentence. (3 pts each) a. r ^ q → p b. p ̅ → q c. q → p ̅ v r 2. Write T if the above item is true and F if it false. Show solution. (3 pts each) a. b. c. C. Show whether propositional or not p ↔ q ≡ (p → q) ^ (q → p) (8 pts) D. Find the converse, inverse and contrapositive of the implication: “If today is Monday then, I have an exam today.” (3 pts each) 1. Inverse: 2. Converse: 3. Contrapositive:

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find a) A ∩ B ∩ C. b) A ∪ B ∪ C. c) (A ∪ B) ∩ C. d) (A ∩ B) ∪ C.

Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (B − A) ∪ (C − A) = (B ∪ C) − A.

Find the truth set of each of these predicates where the domain is the set of integers. a) P(x): 𝑥 3 ≥ 1 b) Q(x): 𝑥 3 = 8 c) R(x): x < 𝑥 2 d) F(x): |x +5| < 52

What is the cardinality of each of these sets? a) {a, 0, {a, 0}} b) {{a}} c) {∅, a, {a}} d) {0, 1, a, {a}, {a, {a}}}

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}

Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.

Determine whether these statements are true or false. a) ∅ ∈ {∅} b) ∅ ∈ {∅, {∅}} c) {∅} ∈ {∅} d) {∅} ∈ {{∅}} e) {∅} ⊂ {∅, {∅}} f ) {{∅}} ⊂ {∅, {∅}} g) {{∅}} ⊂ {{∅}, {∅}}

Let A = {m, n, o}, B = {a, b}, and C = {7, 2}. Find: a) A x B x C. b) C x B x A. c) C x A x B. d) B x B x B.Let A = {m, n, o}, B = {a, b}, and C = {7, 2}. Find: a) A x B x C. b) C x B x A. c) C x A x B. d) B x B x B.

Draw De Morgan’s Law, Absorption and Complementation Laws

If n fair six-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n  4 ?

We are given two functions f : A → B and g : B → C. Prove that if f and g are onto, then g ◦ f is onto.

For each of the following relations, determine whether they are reflexive, symmetric, anti- symmetric, and/or transitive, and give a brief justification for each property. a) R ⊆ Z × Z where xRy iff x = y 2 b) The empty relation: R ⊆ …

Let A={1,2,3,4,5} and R be the relation on A such that R={(1,1),(1,4),(2,2),(3,4),(3,5),(4,1),(5,2),(5,5)}. Find the transitive closure of R by Warshall's Algorithm

Let A={1, 2, 3, 4} and R be the relation on A such that R = { (1, 2), (1, 4), (2, 1), (2, 3), (3, 1) }. Find the transitive closure of R by warshall's algorithm.

If n fair six-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n >= 4

Prove that n*P( n -1,n - 1) = P (n, n)

Show that C (n+1, k) = C (n, k -1) + C (n, k)

Prove that using proof by contradiction. √2 + √6 < √15

Use a truth table to verify this De Morgan’s law: ¬(p ∧ q) ≡ ¬p ∨ ¬q

Determine whether each set below is a function from X = {1, 2, 3, 4} to Y = {a, b, c, d}. If it is a function, find its domain and range, draw its arrow diagram, and determine if it is oneto-one (Injective), onto (Surjective), or both…

Determine whether this function f(n) = n2 − 1 is one-to-one, onto, or both. Explain your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers

Let f be the function from x ={0, 1, 2, 3, 4, 5} to X defined by f(x) = 4x mod 6. Write f as a set of ordered pairs and draw the arrow diagram of f . Is f one-to-one? Is f onto?

a) Simplify the following Boolean expression using three variable maps F(x, y, z) = x y z + x’y’ z + x y’ z’ b)Simplify the following Boolean expression using four variable maps: F(w,x, y, z) = Σ(0,1,4,5,6,7,8,9)

Q no.1) Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Draw the Venn diagrams for each of these combinations of the sets A, B, and C. a) A ∩ (B ∪ C) b) A ∩ B ∩ C

Exercise # 3 a. Ten persons have first names Alice, Billy, and Charlie and last names Lee, Mendoza and Navarro. Show that at least two persons have the same first and last names. b. In any group of 367 people, at least two people must…

Exercise # 2 A committee composing of six members naming Ana, Ben, Cora, Dina, Elphie and Francis is to select a chairperson, secretary and a treasurer. d. In how many ways can they select the chairperson, secretary and treasurer? e. …

1. Find the sets A and B if A − B = {1, 5, 7, 8}, B − A = {2, 10}, and A ∩ B = {3, 6, 9}. 2. If B and C are two sets, prove or disprove the identity, B×C=C×B. 3. Prove that if A and B are sets, then A ∩ (A ∪ B) = A. 4. Suppose that A …

Determine whether each of these functions is a bijection from R to R. a)f (x) = x^3

Using mathematical induction, prove that for every positive integer n, 1 · 2 + 2 · 3 + · · · + n(n + 1) = (n(n + 1)(n + 2)) / 3

Let fn be the nth Fibonacci number. Prove that, for n > 0 [Hint: use strong induction]: fn = 1/√5 [((1+√5)/2)n - ((1-√5)/2)n ]

There is a long line of eager children outside of your house for trick-or-treating, and with good reason! Word has gotten around that you will give out 3k pieces of candy to the kth trick-or-treater to arrive. Children love you, denti…

Let the sequence Tn be defined by T1 = T2 = T3 = 1 and Tn = Tn-1 + Tn-2 + Tn-3 for n ≥ 4. Use induction to prove that Tn < 2n for n ≥ 4

For the following recursive functions, Prove that am,n = m · n. Here am,n is defined recursively for (m, n) ∈ N × N. am.n = { 0 if m=0 n+am-1,n if m!=0

Exercise #1: Restaurant Menu Appetizers Nachos …………………… 120.00 Salad ……………………… 125.00 Main Courses Hamburger ………………. 125.00 Cheeseburger …………… 130.00 Fish Filet …………………. 155.00 Beverages Tea ……………………….. 120.00 Milk ………………………. .120.00 …

N=2^{12}=4096N=2 12 =4096 ways How to review the solution in polya strategy

Show, by giving a proof by contradiction, that if 100 balls are placed in nine boxes, some box contains 12 or more balls.

Show by giving a proof by contrapositive, that if 3n+2 is odd, then n is odd

a n+1 −a n =3n characteristic equation: 1/x-1/x^2=01/x−1/x 2 =0 x-1=0x−1=0 x=1x=1 homogeneous solution: a_h=cx^n=ca h =cx n =c particular solution: a_t=An^2+Bn+Ca t =An 2 +Bn+C A(n+1)^2+B(n+1)+C-A…

Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is either sno…

Find ⋃ 𝑨𝒊 ∞ 𝒊=𝟏 and ⋂ 𝑨𝒊 ∞ 𝒊=𝟏 where: 𝑨𝒊 = {𝒊,𝒊 + 𝟏,𝒊 + 𝟐, … } for every positive integer 𝒊.

Given that n is a positive integer and in the expansion of (1+ax)n, the coefficient of x is 200. Find the coefficient of x in the expansion of (1+ax)2n.

Q1 Let R = {(1,4),(2,1),(2,5),(2,4),(4,3),(5,3),(3,2)}. Use warshall’s algorithm to find the matrix of transitive closure where A = {1, 2, 3, 4, 5}

A class has 175 students. The following data shows the number of students taking one or more subjects. Mathematics 100, Physics 70, Chemistry 40; Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23; Math…

Let 𝐴={1,2,3,4}. Determine the truth value of each statement: i. ∀𝑥 ∈𝐴,𝑥+3<6

i. false: 4+3>64+3>6 ii. true: 1+3<61+3<6 iii. false: 2\cdot1^2+1=32⋅1 2 +1=3 2\cdot2^2+2=102⋅2 2 +2=10 2\cdot3^2+3=212⋅3 2 +3=21 2\cdot4^2+4=362⋅4 2 +4=36

Let 𝐴={1,2,3,4}. Determine the truth value of each statement: i. ∀𝑥 ∈𝐴,𝑥+3<6 ii. ∃𝑥,𝑥+3<6 iii. ∃𝑥,2𝑥2 +𝑥 =15

Let Q(x)Q(x) be the statement “x + 1 > 2xx+1>2x ”. If the domain consists of all integers, let us find the following truth values. a) Since 1>0,1>0, we get that 𝑄(0)Q(0) is true. b) Taking into account that it is not true that 0…

Let Q(x) be the statement “x + 1 > 2x.” If the domain consists of all integers, what are these truth values? (7 pts.) a) 𝑄(0) b) 𝑄(−1) c) 𝑄(1) d) ∃𝑥𝑄(𝑥) e) ∀𝑥𝑄(𝑥) f) ∃𝑥¬𝑄(𝑥) g) ∀𝑥¬𝑄(𝑥)

20. Determine the truth value of each statement if the domain consists of all real numbers. (4 pts.) a) ∃𝑥(𝑥3 = −1) b) ∃𝑥(𝑥4 < 𝑥2) c) ∀𝑥((−𝑥)2 = 𝑥2) d) ∀𝑥(2𝑥 > 𝑥)

prove that n.P(n-1,n-1)=p(n,n)

Let R = {(1,4), (2,1), (2,5),(2,4),(4,3),(5,3),(3,2)} on the set A = {1, 2, 3, 4, 5}. Use Warshall’s algorithm to find transitive closure of R.

show that C(n+1,k)=C(n,k-1)+C(n,k)

Let f be the function from x ={0, 1, 2, 3, 4, 5} to X defined by f(x) = 4x mod 6. Write f as a set of ordered pairs and draw the arrow diagram of f . Is f one-to-one? Is f onto?

Which of the following functions are injective? Which are surjective? a) f: Z → Z given by f(x) = x2 + 1. b) g: N → N given by g(x) = 2x. c) h: R → R given by h(x) = 5x - 1.

If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.

Create a schematic diagram of all odd numbers from 20 to 45

(a) If x is nonnegative, then x is positive or x is 0.

4. v defined by vn = n! + 2, n ≥ 1. (a) Find v3 (b) Find Σ 4 on top , i=1 at the bottom and Vi on the right hand side of Sigma (c) Is v increasing, decreasing, non-increasing or non-decreasing?

1.b is defined by bn = n(−1)n, n ≥ 1 (a) Find Σ when 4 is on top, i=1 at the bottom and bi at the right hand side (b) Is b increasing, decreasing, non-increasing or non-decreasing?

Given ϕ2 = 5. Is ϕ increasing, decreasing, non-increasing and/or non-decreasing?

Find all substrings of the string babc.

1.Draw the digraph of the following relations. (a) R = {(1, 2), (2, 3), (3, 4), (4, 1)} on {1, 2, 3, 4} (b) R={(1, 2), (2, 1), (3, 3), (1, 1), (2, 2)} on X = {1, 2, 3}

For each relation below, determine if they are reflexive, symmetric, anti-symmetric, and transitive. (a) X= { 1, 2, 3, 4} R1={(1, 2),(2, 3),(3, 4)} (b) X = {a, b, c, d, e} R1 = { (a, a), (a, b), (a, e), (b, b), (b, e), (c, c),…

A discrete mathematics class contains I mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores. 2 mathematics majors who are juniors. 2 computer science majors …

Use quantifiers and predicates with more than one vari- able to express these statements. a) There is a student in this class who can speak Hindi. b) Every student in this class plays some sport. c) Some student in this class has …

Create a schematic diagram of all odd numbers from 20 to 45

State whether the following graph g1 and g2 are isotropic or not

Find the conjunction of the propositions p and q where p is the proposition “Rebecca’s PC has more than 16 GB free hard disk space” and q is the proposition “The processor in Rebecca’s PC runs faster than 1 GHz.”

all swedish movies are serious

For each of the following compound propositions give its truth table and derive an equivalent compound proposition in disjunctive normal formal (DNF) and in conjunctive normal form (CNF). (a) (p → q) → r (b) (p ∧…

For each of the given statements: 1 - Express each of the statements using quantifiers and propositional functions. 2 - Form the negation of the statement so that no negation is to the left of the quantifier. 3 - Express the …

Translate these system specifications into English where the predicate S(x, y) is “x is in state y” and where the domain for x and y consists of all systems and all possible states, respectively. (a) ∃S(x, open) (b) ∀x(S(x…

(a) Find the inverse of 19 modulo 141, using the Extended Euclidean Algorithm. Show your steps.

A says “If B is a knight, then I am a knave”, B says nothing.

(e) Propositions 𝑝 and 𝑞 are defined on the universal set 𝑈 = {𝑞𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙𝑠}. Let 𝑝 be the proposition “𝑥 is a square” and 𝑞 be the proposition “𝑥 is a rectangle”. If the implication is given as 𝑞 → 𝑝 , then state the conve…

Taking the long view on your education, you go to Econet Wireless Network Pvt Ltd and ask what you should do in university to be hired when you graduate. The personnel director replies that you will be hired only if you major in mathe…

Find the simplest form of given Boolean expressions using algebraic methods. i. A(A+B) + B(B+C) + C(C+A) ii. (A+|B|)(B+C) + (A+B)(C+|A|) iii. (A+B)(AC+A|C|)_+AB +B iv. |A|(A+B) + (B+A)(A+|B|)

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…

Detarmine whether each of these functions is a bijection from R to R

Find these values ⌊−3.1⌋

Detarmine whether each of these functions is a bijection from R to R

Let A= {1, 2, 3, 4}, and let R And S be the relations on A As follow: R = {(1,1),(1, 4),(3,2),(4,2),(4,3)} S = { (1,1),(1,2),(2,2),(3,3),(4,2),(4,4)) Then find out: 1. M𝑅𝑐 2. 𝑀𝑠̅ 3. M (𝑅∪ S)

The options available on a particular model of a car are five interior colors, six exterior colors, two types of seats, three types of engines, and three types of radios. How many different possibilities are available to the consumer?

How many different car license plates can be constructed if the licenses contain three letters followed by two digits if repetitions are allowed? If repetitions are not allowed?

In how many ways can a string of length 5 be formed using the letters ABCDEFG without repetitions. It should begin with the letter F and end with the letter A.

Three departmental committees have 6, 12, and 9 members with no overlapping membership. In how many ways can these committees send one member to meet with the president? Use the addition principle.

In how many ways can a diner choose one item from among the appetizers and main courses at Kay’s Quick Lunch. Refer to Figure 1. Use the addition principle.

Two dice are rolled. How many outcomes give the sum of 7 or the sum of 11?

In how many ways can we select a chairperson, vice- chairperson, and recorder from a group of 11 persons?

At one point in the Illinois state lottery Lotto game, a person was required to choose six numbers (in any order) among 44 numbers. In how many ways can this be done?

Suppose that a pizza parlor features four specialty pizzas and pizzas with three or fewer unique toppings (no choosing anchovies twice!) chosen from 17 available toppings. How many different pizzas are there?

How many number of strings that can be formed by ordering the letters given: (a) SCHOOL (b) CLASSICS

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Question ID: mtid-5-stid-8-sqid-3169-qpid-1868