Solved: State the converse, inverse, and contrapositive of each of the given statements. If it snows tonight, I will stay home. I go to the beach whenever it is a sunny summer day. A positive integer is prime only if it has no divisors other than 1 and itself.

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

There are piles of identical red, blue, and green balls where each pile contains at least 10 balls. (a) In how many ways can 10 balls be selected if exactly one red ball must be selected? ( (b) In how many ways can 10 balls be selec…

There are a selection of Action Comics, Superman, Captain Marvel, Archie, X-Man, and Nancy comics. (a) How many ways are there to select 10 comics? (b) How many ways are there to select 10 comics if we choose at least one of each bo…

Refer to the menu below, how many dinners at Kay’s Quick Lunch consist of an optional appetizer, one main course, and an optional beverage? Appetizers Nachos 2.15 Salad 1.90 Main courses Hamburger 3.25 Cheeseburger 3.65 Fi…

Two fair dice are rolled. (a) What is the probability of getting a sum of 7? (b) What is the probability that the sum of the numbers on the dice is odd?

There are 5 green 7 red balls. Two balls are selected one by one without replacement. Find the probability that first is green and second is red.

In a hamster breeder's experience the number X of live pups in a litter has the probability distribution of x 3 4 5 6 7 8 9 P(x) 0.04 0.1 0.26 0.31 0.22 0.05 0.02 (a) Find the probability that the next litter will produce five …

45% of the children in a school have a dog, 30% have a cat, and 18% have a dog and a cat. What percentage of those who have a cat also have a dog?

When drawing one card out of a deck of 52 playing cards, what is the probability of getting heart or a face card? There are 13 hearts and a total of 12 face cards (3 of each suit: spades, hearts, diamonds and clubs), but only 3 face c…

Draw a graph having the given properties (a) Four vertices each of degree 1 (b) Six vertices each of degree 3

What is 7 mod 3? a) 1 mod 3 b) 2 mod 3 c) 3 mod 3 d) 0 mod 3

What is 6 mod 9 + 11 mod 9 congruent to? a) 8 mod 9 b) 6 mod 9 c) 7 mod 9 d) 0 mod 9

Which of the following expressions are equivalent to: (-171 x 77) mod 16 Select all that applies a) 1 b) (5 x 13) mod 16 c) 2 d) 65 mod 16 e) 18 mod 16

There are 81 groups of 21 students. When we regroup all of the students so that each group has 5 members, how many students will be left without a group?

I do my laundry every 20 days. Last week, laundry day fell on a Friday. On what day of the week will laundry fall next?

Find five different integers a such that a≡3 (mod 6)

Find the value of x if 2x ≡ 7 (mod 11). (*Hint: The value of x should be between 0 to 10)

Perform the following modular arithmetic by applying the addition/subtraction rule. (a) (9+15) mod 7 (b) (15-23) mod 5 (c) (-47 x (5+1)) mod 5

Find a number which is congruent to 2 mod 3 and congruent to 3 mod 4.

It is 8:00 AM in our 24 hour world. What time is it in a 3 hour world?

. Andy is facing West, he rotates 1260 clockwise. What direction is he now facing? (Note: A circle has 360 degrees)

The following word was encrypted using a Caesar cipher with a shift of 2: ecguct. What word is it?

Using Caesar cipher, encrypt the word “pizza” mathematically with a shift of 3

Decrypt the word “xqjjud” mathematically with a shift of 5

Given the following Boolean expression on it, draw a logic gate circuit for this function. AB+ C (A+B)

Complete the truth tables for these two Boolean expressions: (a) Output = A + B (b) Output = A + AB

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)

2. What does it mean for two propositions to be logically equivalent?

5. Let p be the proposition “I will solve every question in this assignment” and q be the proposition “I will be fully prepared for upcoming topics” Express each of these as a combination of p and q. a. I will be fully prepared for up…

3. Give an example of a predicate P(x,y) such that ∃x∀yP(x,y) and ∀y∃xP (x, y) have different truth values.

4. Suppose that a statement of the form ∀xP(x) is false. How can this be proved?

1. Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0). Find the a) reflexive closure of R. b) symmetric closure of R. 2. Find the t…

4. Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3),…

Define a binary relation P from R to R as follows: for all real numbers x and y, (𝑥, 𝑦) ∈ 𝑃 ⇔ 𝑥 = 𝑦^2 . Is P a function? Explain.

1. Consider the K-Maps given below. For each K- Map i. Write the appropriate standard form (SOP/POS) of Boolean expression. ii. Design the circuit using AND, NOT and OR gates. iii. Design the circuit only by using • NAND gates if …

Let R={1,1),(1,2),(2,1), (2,1), (3,3), (4,1), (5,1)} define on set A= {2,3,4,5,6}

(a) Find the inverse of 19 modulo 141, using the Extended Euclidean Algorithm. Show your steps. (b) Solve the congruence 19x ≡ 7 (mod 141), by specifying all the integer solutions x that satisfy the congruence. ≡

(a) Use Fermat’s little theorem to compute: 4101 mod 5, 4101 mod 7, 4101 mod 11. (b) Use your results from part (a) and the Chinese Remainder Theorem to compute 4 101 mod 385. (note that 385 = 5 × 7 × 11).

Encrypt the message ATTACK using the RSA cryptosystem with n = 43 · 59 and e = 13, translating each letter into integers and grouping together pairs of integers, as done in example 11 in the textbook and in the classnotes.

Use the Well Ordering Principle to prove that any integer greater than or equal to 23 can be represented as the sum of nonnegative integer multiples of 6, 7 and 17.

Let α, β be roots of the equation x^2− 3x − 1 = 0. For each nonnegative integer n, let y_n = α^n + β^n . Show that gcd(y_n, y_(n+1)) = 1 for each nonnegative integer n.

Determine an, the number of words of length n on the alphabet {a, b, c} which do not contain the substring ab. For instance, a3 = 21 since there are 21 such words with 3 letters, namely: aaa aac aca acb acc baa bac bba b…

Let α, β be roots of the equation x^2− 3x − 1 = 0. For each nonnegative integer n, let y_n = α^n + β^n . Show that gcd(y_n, y_(n+1)) = 1 for each nonnegative integer n.

Determine an, the number of words of length n on the alphabet {a, b, c} which do not contain the substring ab. For instance, a3 = 21 since there are 21 such words with 3 letters, namely: aaa aac aca acb acc baa bac bba b…

Q1. Let {1,2,3,4,6,9} with the partial order of divisibility. Draw its Hasse Diagram Q2. [{1,2,3,4,5}, s], Draw its Hasse Diagram

[{1,2,3,4,5}, <=], Draw its Hasse Diagram.

State the converse, inverse, and contrapositive of each of the given statements. If it snows tonight, I will stay home. I go to the beach whenever it is a sunny summer day. A positive integer is prime only if it has no divisors other than 1 and itself.

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Question ID: mtid-5-stid-8-sqid-3221-qpid-1920