Solved: Let H(x) be the statement “x is hardworking”, N(x) be the statement “x is naughty” and C(x) be the statement “ x is clever”, where the domain for x consists of all students. Use quantifications to express each of the following statements. a) Some students are clever but naughty. b) Not all students are clever and hardworking. c) Some students are clever, hardworking and not naughty. d) All students are clever, or hardworking or naughty.

Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.

Consider a relation R=\ (1,1),(1, ), (0,2), (2,3) (3,1)) on the set A=\ 1,2,3\ Find transitive closure of the relation R using algorithm Warshall's

Find matrix and digraph of the relation R=\ (x,y)/xRy iff 2+y is every defined on the set A=\ 1,2,3,4\

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation defined by aOb = max{ a,b}.

Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.

Proof that an undirected graph has an even number of vertices of odd degree.

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime implicants, and essential prime implicants of F(x, y, z).

Express the given proposition using P,Q,R loyal connectives P:you are sick Q:you miss tha examination R: you pass this subject You did not miss the examination if and only if you pass the subject

The English alphabet contains 21 consonants and fivevowels. How many strings of six lowercase letters of theEnglish alphabet contain: a) exactly two vowels? b) at least two vowels?

Show that a lattice is distributive if and only if for any elements a,b,c in thee lattice (aVb) V c< aV (bVc)

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c,…

A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found! …

For the relation R = {(p,p) ,(q,p),(q,q),(r,r),(r,s),(s,s) ,(s,m) ,(m,m)} 1.Using warshall algorithm find the transitive closure R* of R 2.write matrix representation of R* 3.Check whether the relation of R* is an equiv…

Consider a relation R=\ (1,1),(1, ), (0,2), (2,3) (3,1)) on the set A=\ 1,2,3\ Find transitive closure of the relation R using algorithm Warshall's

Define a relation - on the set of real numbers by x-ymeans I x | + I y I : I x+y l. which of the properties for an equivalence relation does - satisfy?

Define a relation R on {a,b,c, int i* e . a Reflexive but not symmetric ↳ Symmetric but not transitive <> Transitive but not reflexive

solve the following recurrence relations a. 𝑇(𝑛) = 𝑇( 𝑛/4) + 𝑇( 𝑛/2 ) + 𝑛^2 b. T(n) = T(n/5) + T(4n/5) + n c. 𝑇(𝑛) = 3𝑇( n/4 ) + 𝑐𝑛^2 f. 𝑇(𝑛) = (𝑛/𝑛−5) * 𝑇(𝑛 − 1) + 1 g. 𝑇(𝑛) = 𝑇(log 𝑛) + log 𝑛 h. 𝑇(𝑛) = 𝑇 (𝑛^ 1/ 4) + 1 i. 𝑇(𝑛…

State TRUE or FALSE justifying your answer with proper reason. a. 2𝑛^2 + 1 = 𝑂(𝑛^2 ) b. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 ) c. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 log 𝑛) d. 3𝑛^2 + √𝑛 = 𝑂(𝑛 + 𝑛√𝑛 + √𝑛) e. √𝑛 log 𝑛 = 𝑂(𝑛)

the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive steps of Warshall’s Algorithm: Step 1: Execute W:=M_R, k:=0W:=M R ,k:=0 Step 2: Execute k:=k+1k:=k+1 S…

Consider a relation R= {(1, 1) (1, 3), (2, 2), (2,3) (3,123 on the set A = {1,2,37 Find transitive using warshalls algorithm... closure of the relation R consider

Consider a relation R= {(1, 1) (1, 3), (2, 2), (2,3) (3,1)on the set A = (1,2,3) Find transitive using warshalls algorithm... closure of the relation R consider

Consider two sets A and B such that A ⊆ B. Find the possible values of x if A = {2, 4, 6, x} and B = {2, 3, 5, 6, x+1}. Ans: x = 3. Please explain.

1. The following formulas have been abbreviated based on the common abbreviation rules. Follow the steps below and translate the formulas into good English. · Step 1: Re-add the omitted brackets. · Step 2: If necessa…

As for those three guys, John, Bill and Bob, I will only vote for one. Translate into proposition logic.

) How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other?

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c,…

Determine whether f : Z × Z → Z is onto if f(m,n)= m-n

Let R={(1,2),(2,1),(2,3),(3,4),(4,1) } be a relation on the A={1,2,3,4}. Find the transitive closure of R using Warshall’s algorithm.

Let A consist of all 52 cards in an ordinary deck of playing cards. Suppose that this deck is shuffled and a hand of five cards is dealt. A list of cards in this hand, in the order in which they were dealt, is a permutation of A…

How many pairs of dance partners can be selected from a group of 12 women and 20 men

There are 7! = 5040 ways in which seven people can form a line. In how many ways can seven people form a circle

In how many ways can the letters of the English alphabet be arranged so that there are exactly ten letters between a and z?

How many different sequences, each of length r, can be formed using elements from A if (a) elements in the sequence may be repeated? (b) all elements in the sequence must be distinct? Expert's answer

A man, a woman, a boy, a girl, a dog, and a cat are walking down a long and winding road one after the other. (a) In how many ways can this happen? (b) In how many ways can this happen if the dog comes first? (c) In how ma…

Total number of stalls for animals in exhibition = 10 Number of each of three type animal = at least 10 Types of animals = 3 (animals; lion, pussycat) Then, possible number of ways of arranging the exhibition are: 3^{10}=3\…

There are 10 stalls for animals in an exhibition. Three animals; lion, pussycat and horse are to be exhibited. Animals of each kind are not less than 10 in number. What is the possible number of ways of arranging the exhibition?

In how many ways can 10 different examination papers be scheduled so that (i) the best and the worst always come together? (ii) the best and the worst never come together?

Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use wh…

Wandana is going to toss a coin eight times. In how many ways can she get five heads and three tails

Wandana is going to toss a coin eight times. In how many ways can she get five heads and three tails Expert's answer

In how many ways can 20 students out of a class of 32 be chosen to attend class on a late Thursday afternoon (and take notes for the others) if (a) Paul refuses to go to class? (b) Michelle insists on going? (c) Jim and Mi…

group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission (assuming that all crew members have the same job)? Expert's an…

A person has 8 children of them he takes 3 at a time to a circus. He does not take the same three children twice to the circus. How many times he will have to go to circus to ensure that every three children have seen the circus…

How many students are offering both biology and chemistry,if only 70 students are offering neither biology or chemistry.

Define and what Cryptography in mathematical foundations of computer science

show that ~Q,P—>Q=>~P in mathematical foundations of computer science

How many 5 vertices unlabelled trees are there? How would I draw them?

Labelled and Unlabelled trees. (a) How many labelled trees of order 5 are there? (b) Draw all unlabeled tree of order 5 (under isomorphism). Hint: make cases on the diameter size.

15.Determine whether f: R to R, defined as f (x) = −3x + 4 is a bijection. Is f invertible, and if it is, what is its inverse? 16.Find the inverse of f (x) = 𝑥+1 𝑥+2 , on a suitable subset of R.

solve the following recurrence relations a. 𝑇(𝑛) = 𝑇( 𝑛/4) + 𝑇( 𝑛/2 ) + 𝑛^2 b. T(n) = T(n/5) + T(4n/5) + n c. 𝑇(𝑛) = 3𝑇( n/4 ) + 𝑐𝑛^2 f. 𝑇(𝑛) = (𝑛/𝑛−5) * 𝑇(𝑛 − 1) + 1 g. 𝑇(𝑛) = 𝑇(log 𝑛) + log 𝑛 h. 𝑇(𝑛) = 𝑇 (𝑛^ 1/ 4) …

for example: p\lor q:p∨q: John wants to come to the class or John will come to the class today r\land s:r∧s: John audits the class and John is enrolled in the class

1. The following formulas have been abbreviated based on the common abbreviation rules. Follow the steps below and translate the formulas into good English. · Step 1: Re-add the omitted brackets. · Step 2: If necessa…

Let A, B, C, D denote, respectively, art, biology, chemistry, and drama courses. Find the number N of students in a dormitory given the data: 12 take A, 5 takeAand B, 4 takeB and D, 2 take B, C,D, 20 take B, 7 takeAand C, 3 t…

LetR={(1,2),(2,1),(2,3),(2,4),(4,1),(4,3)} R={(1,2),(2,1),(2,3),(2,4),(4,1),(4,3)} is a relation on setA={1,2,3,4} Suppose aRnb means that there is a path of length n from a to b Which of elements are of R∞

Let R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} is a relation on set A={1,2,3,4} A={1,2,3,4} Suppose a Rn b means that there is a path of length n from a to b Which of the elements are …

R= {(1,3) ,(1,4) , (3,2) , (3,3), (3,4)} on A={1,2,3,4}

List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ∈ R if and only if a | b.

write a logical conclusion of the following sets of statements (1)if david gets an A in math class, he will receive money from his parents (2)if david receives money from his parents he will save it up (3)if david saves his money, he …

R= {(1,3) ,(1,4) , (3,2) , (3,3), (3,4)} on A={1,2,3,4}

[T] Find \mathbb{Q}\cup \mathbb{N}Q∪N ?

Show using the rules of resolution/inference, that no single assignment of truth values to p, q, r makes all the disjunctions p V-9, p V-9,9 V r, qVT, V revaluate to true. Proof using truth na

multiple-choice test contains 10 questions. There are four possible answers for each question. a) In how many ways can a student answer the questions on the test if the student answers every question? b) In how many ways can a student…

Out of a group of ten residents in a certain county, 3 are Republicans, 5 are Democrats, and 2 are Independents. How many unique partitions of this group of residents are there by political party? 38) You have eight distinct …

Use the binomial theorem to find the coefficient of x^a y^b in the expansion of (2x^3 − 4y^2) 7, where a) a = 9, b = 8. b) a = 8, b = 9. c) a = 0, b = 14. d) a = 12, b = 6. e) a = 18, b…

Find the expansion of a) (x + y)^6 b) (x + y)^4 31) What is the coefficient of x^12 y^13 in the expansion of (x + y)^25? 32) What is the coefficient of x9 in (2 − x)19? 33) What is the coefficient of x^101 y^99 in the e…

bit is either 0 or 1: a byte is a sequence of 8 bits. Find (a) the number of bytes that can be formed (b) the number of bytes that begin with 11 and end with 11 (c) the number of bytes that begin with 11 and do not end with 1…

a) In how many ways can a committee of 5 be chosen from 9 people? (b) How many committees of 5 or more can be chosen from 9 people? (c) In how many ways can a committee of 5 teachers and 4 students be chosen from 9 teac…

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that a) A ∪ B = B ∪ A. b) A ∩ B = B ∩ A.

How many pairs of dance partners can be selected from a group of 12 women and 20 men?

Let R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} is a relation on set A={1,2,3,4} A={1,2,3,4} Suppose a Rn b means that there is a path of length n from a to b Which of the elements are …

Answer the following items. Show your complete answer on a separate sheet of paper. Prove that the following sentences are tautologies. 1. p →p 2. p → (p V q) 3. [p Λ (p → q)] → q 4. p V ~p 5. q → (p V ~p) 6. ~p…

Show that the relation R = ∅ on a nonempty set S is symmetric and transitive but not reflexive.

Coding theory in mathematical foundations of computer science

What and Define The Hamming Metric with example in mathematical foundations of computer science

Theorem 2:- An (m,n) encoding ∅ corrects K or fewer errors if and only if the minimum distance between encoded words is atleast 2K+1 Proof:- Suppose that ∅ corrects K or fewer errors x is an encoded word of distance 1≤j≤k …

One hundred students were asked whether they had taken course in any of the three areas, CSE, ME and EE. The results were. 45 had taken CSE, 38 had taken ME, 21 had taken EE, 18 had taken CSE and ME, 9 had taken CSE and…

Consider a Boolean expression Ex,y,z=xzv(y^z). Find disjunctive and conjunctive normal form.

For each stall there are 3 ways to exhibit one of three animals. Since there are 10 stalls, by Combinatorial Product Rule, the number of ways of arranging the exhibition is equal to 3\cdot 3\cdot\ldots\cdot 3=3^{10…

There are 10 stalls for animals in an exhibition. Three animals; lion, pussycat and horse are to be exhibited. Animals of each kind are not less than 10 in number. What is the possible number of ways of arranging the exh…

Show that the following logical equivalences hold for the Peirce arrow ↓, where P ↓ Q ≡ ∼(P ∨ Q). a. ∼P ≡ P ↓ P b. P ∨ Q ≡ (P ↓ Q) ↓ (P ↓ Q) c. P ∧ Q ≡ (P ↓ P) ↓ (Q ↓ Q) H d. Write P → Q using Peirce arrows on…

Obtain the Conjunctive Normal Form of (x^y) V (-x^y)

Define and give examples of injective surjective and bijective functions. Check the injectivity and surjectivity of the following function f: NN given by f(x)=x2

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Question ID: mtid-5-stid-8-sqid-3770-qpid-2469