Solved: Suppose that a truth table in n propositional variables is specified. Show that a compound proposition with this truth table can be formed by taking the disjunction of conjunctions of the variables or their negations, with one conjunction included for each combination of values for which the compound proposition is true. The resulting compound proposition is said to be in disjunctive normal form.

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.

If A = {1, 2, 3, 4, 6, 12} then define a relation R by aRb if and only if a divides b. Prove that R is a partial order on A.

Find the minimum number n of integers to be selected from S = {1, 2, . . . , 9} so that: (a) The sum of two of the n integers is even. (b) The difference of two of the n integers is 5.

Consider a tournament with n players where each player plays against every other player. Suppose each player wins at least once. Show that at least 2 of the players have the same number of wins.

Suppose 5 points are chosen at random in the interior of an equilateral triangle T where each side has length two inches. Show that the distance between two of the points must be less than one inch.

Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.

There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed?

In a survey on the gelato preferences of college students, the following data was obtained: • 78 like mixed berry • 32 like irish cream • 57 like tiramisu • 13 like both mixed berry and irish cream • 21 like both irish cream and…

Two Balls are to be selected without replacement from a bag that contains one red, one blue, one green and one orange ball. A) Use the counting principle to determine the number of possible points in the sample space. Construct …

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

In a mathematics contest with three problems, 80% of the participants solved the first problem, 75% solved the second and 70% solved the third. Prove that at least 25% of the participants solved all three problems.

Suppose that in a bushel of 100 apples there are 20 that have worms in them and 15 that have bruises. Only those apples with neither worms nor bruises can be sold. If there are 10 bruised apples that have worms in them, how many…

Suppose among 32 people who save paper or bottles (or both) for recycling, there are 30 who save paper and 14 who save bottles. Find the number m of people who: (a) save both; (b) save only paper; (c) save only bottles.

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…

Two Balls are to be selected without replacement from a bag that contains one red, one blue, one green and one orange ball. A) Use the counting principle to determine the number of possible points in the sample space. Construct …

Mark and Erik are to play a tennis tournament. The first person to win two games in a row or who wins a total of three games wins the tournament. Find the number of ways the tournament can occur.

Teams A and B play in a tournament. The first team to win three games wins the tournament. Find the number n of possible ways the tournament can occur.

There are four roads from city X to Y and five roads from city Y to Z, find (i) how many ways is it possible to travel from city X to city Z via city Y. (ii) how different round trip routes are there from city X to Y to Z to Y a…

Three persons enter into car, where there are 5 seats. In how many ways can they take up their seats?

1. Let p and q be the propositions “Swimming at the Corregidor Island shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence.

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,…

Let A = {2, 3, 5, 7, 11, 13} and B = {A, 2, 11, 18}. 1. Find A ∪ B. 2. Find A ∩ B. 3. Find A − B.

Let A = {x ∈ R : x 2 = 2} and B = {x ∈ R : x ≥ 0}. 1. Find A ∩ B. 2. Find A ∪ B. 3. Find A − B. 4. For U = R, find Ac and Bc . 5. Find N − B.

There are 21 pupils in a grade 7 class. The class teacher has to choose eight of the pupils for a group that will visit Germany in three months time. In how many different ways can the teacher select pupils for the group ?

If the truth value of p is "false", the truth value of q is "false", and the truth value of r is "true", which of the following expressions is "false"?

Determine whether each of these functions from {a,b,c,d} to itself is one-to-one. a) f(a)=b, f(b)=a, f(c)=c, f(d)=d

State whether the null hypothesis should be rejected on the basis of given P-value. P-value 0.002, a 0.01, one-tailed test.

Find the solution of the recurrence relation an = 4an−1 − 3an−2 + 2 n + n + 3 , a0 = 1 and a1 = 4

Determine the cardinality of the power set of the set {2,3,x,y,z}

Let={2,4,7},B={2,5,4} find the set A∆B

Let A = { y € Z | y = 10b - 3 for some integer b } B = { z € Z | z = 10c + 7 for some integer c } Prove or Disprove that 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.

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.

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Question ID: mtid-5-stid-8-sqid-3744-qpid-2443