Solved: Q1 Find the floor and ceiling Function of x and . Take any value of x and y. (5) X Floor Ceiling -X.Y (x point y) X / Y (x divided by y)

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

A school organized a book fair and in this book fair a book seller is selling his books under the following rules: There are three different packages available. First package contains 2 Islamic books, 2 Science books and 2 Geography…

The argument is p→~q,~r→p,q|–r in true table in mathematical foundations of computer science

How many license plates can be made using either two or three letters followed by either two or three digits?

Show ((p ∨ q) ∧ ¬(¬p ∧ (¬q ∨ ¬r))) ∨ (¬p ∧ ¬q) ∨ (¬p ∧ ¬r) is tautology, by using replacement process.

I come to class whenever there is going to be a quiz this statement is inverse converse and contarpositve?

An engineer designs at least one robot a day for 30 days. If a total of 45 robots have been designed, then show that there must have been a series of consecutive days when exactly 14 robots were designed.

Let an denote the number of surjective (onto) functions f : {1, 2, . . . , n} −→ {1, 2, 3} such that f(1) < f(2). Give a Θ estimate for an.

A committee of 8 is to be formed from 16 men and 10 women. In how many ways can the committee be formed if i) there are no restrictions. ii) there must be 4 men and 4 women. iii) there should be an …

A committee of 8 is to be formed from 16 men and 10 women. In how many ways can the committee be formed if i) there are no restrictions. ii) there must be 4 men and 4 women. iii) there should be an even number of women. …

solve the recurrence t(n)=(t(n/2)^2) assuming t(1)=1

list the ordered pairs in the equivalence relations R induced by these partitions of p { {1} , {3} , { 2,4,5,6} rt he set of { 1,2,3,4,5,6}

Suppose there are 10 male and 6 female professors to teach Discrete mathematics. In how many ways a student can choose Discrete mathematics professor.

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 …

The argument is p→~q,~r→p,q|–r in true table in mathematical foundations of computer science

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

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,…

Draw a simple, undirected graph yourself, the vertices are connected with each other including 8 vertices and 14 edges. Find the shortest path from two arbitrary vertices: a) The weight of each edge is 1. b) Self-weighting for…

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…

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 necessary, con…

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)

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…

How many positive integers less than 100 is not a factor of 2,3 and 5?

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…

Write the converse, inverse, and contrapositive of the following conditional propositions. (Hint: If applicable, write each conditional proposition in standard form first.) a. Rose may graduate if she has 120 hours of OJT credits. …

How many different plates are there that involve 1, 2 or 3 letters followed by 1, 2, 3 or 4 digits? How many 2 digit or 3-digit numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed?

The following table shows the income distribution of 600 families. Find the minimum income of the riches 30% families. Also the limits of income of middle 50% of families, to the nearest rupees. Income Below 75 75- 1…

a) Suppose P (x, y) denotes the equation8, what will the truth values of the Propositions P (2,2), P (0,4).

Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

Let 8Z be the set of all integers that are multiples of 8. Prove that 8Z has the same cardinality as 3Z, the set of all integers multiples of 3.

We assume the question is: What is mathematical foundation for computer science? Solution: The mathematical side concentrates on areas where computers are used, or which are relevant to computer science, namely algebra, general topo…

Assignment problems in mathematics foundations of computer science

What and Define arrow diagram in mathematical foundations of computer science

Q1 Find the floor and ceiling Function of x and . Take any value of x and y. (5) X Floor Ceiling -X.Y (x point y) X / Y (x divided by y)

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