Solved: Construct a proof for the five color theorem for every planar graph. 2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as mileage of the vehicle, etc.) as much as you can. Essentially consider the two fold, - Routes with the shortest distance (Quick route travelling by own vehicle) - Route with the lowest cost

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

Which of the following is false about a proposition? (¬p ∨ p) is a proposition.(¬p ∧ p) is a proposition.A predicate is not a proposition.c.(¬p ∨ ¬p) is a proposition.

Translate the following statement into mathematical expression based on the given propositional variables. “If a country’s leaders are elected based on good leadership traits, then the country will have good governance”. p : A count…

Calculate the quartiles, (i,e., Q1 and Q3,), deciles D3 and D7, Percentiles P10, P23 and P92 for the following data. Also find the Quartile Range and Quartile Deviation. Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No…

The ages of 100 persons are tabulated below. Find Q1, Q3, D2, D8, P20 and P60. Above what age we have the oldest 10% persons? Age(in years) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of persons 16 20 21 28 10 3 2 Note: …

What is symbolic form of it is not true that, the students answered their activity or participated in the class discussion

what is symbolic form of she did nit received a promotion or she did not received a salary increase

3Fn − Fn−2 = Fn+2, for n ≥ 3.

2. A survey on subjects being taken by 250 college students in Candon City revealed the following information. Subject Number of students taking the subject Mathematics 145 Filipino 90 English 88 Math and Filipino 25 Filipin…

Produce truth tables for given Boolean expressions A’B’C + AB’C + ABC + ABC’

Consider the recurrence relation an= 3 an1 + 3 an-2 when ao = 1 and a1= 2 then a3= ..: O 56

A student can choose a computer project from one of three lists. The three lists contain 23, 15 and 19 possible projects, respectively. How many possible projects are there to choose from?

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

Let S = [1,2,4,5,10,20, 25, 50, 100). Then show that 5 forms a lattice under divisibility. Draw the Hasse diagram also.

Suppose we are given a Boolean function f(x, y, z)=(x+y)(x+y)(x+2). Find its DN form

Determine how many bit strings of length 6 can be formed, where the last bit is 0 and two consecutive 0s are not allowed.

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

Consider the sequence (wn) defined recursively by w2 = 3 wn = (wn-1 (1 - 0.02)) - 2(n + 1)/ n2 for n >= 3. Write a MATLAB program to compute wn for n = 2, 3, 4, . . . , 10 and display the values n and wn …

Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}

Let f : A → B, g : B → C, and h : C → D be functions. 1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13 2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal? 3. U…

The chairs of an auditorium are to be labeled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently?

If 2 distinguishable dice are rolled, in how many ways can they fall? If 5 distinguishable dice are rolled, how many possible outcomes are there? How many if 100 distinguishable dice are tossed?

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?

How many numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed? 13) Suppose that we draw a card from a deck of 52 cards and replace it before the next draw. In how many ways can 10 ca…

In how many ways can 10 people be seated in a row so that a certain pair of them are not next to each other? Three persons enter into car, where there are 5 seats. In how many ways can they take up their seats?

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

Exercise 9: Draw a full binary tree having the following properties 1. Four internal vertices and five terminal vertices. 2. Height = 3 and nine terminal vertices. 3. Height = 4 and nine terminal vertices.

Use any of the two proof methods to prove: ((~a^b)^(b^c))^~b

((~a^b)^(b^c)^~b

let q(x) denote the statement x is an integer . what are the truth values of the following?

If the statement q ^ r is true, determine all combinations of truth values for p and s such that the statement (q -> [¬p v s]) ^ [¬s -> r] is true.

a) Suppose P (x, y) denotes the equation y=x+8, what will the truth values of the Propositions P (2,2), P (0,4).

Ask about the lenght of 5 formed using the lettre ABCDEFG without repetitions. How many string begins with the letter F and end with the letter A

For each recurrence relation and initial conditions, find: (i) general solution; (ii) unique solution with the given initial conditions: (a) an = 3an−1 + 10an−2; a0 = 5, a1 = 11

If A= {1,3,4,6,7} B= {1,2,3,7,6,9,4} C={3,2,7,9,6,1,5,8,4} depict sets A,B and C in a Venn diagram

If x and y are not both even numbers, then neither the square of x and y is odd, nor the cube of x and y is even

Let R be a binary relation on N × N defined by (w, x ) R (y, z) if and only if w = y and x ≤ z . Is R reflexive? Is R symmetric? Is R antisymmetric? Is R transitive?

For each of the following sets, a) S={1,2,3}, T ={a, b, c} b) S={a, b}, T ={1,2,3,4} c) S={1,2,3,4}, T ={a, b} Determine whether 1. There is a one-to-one function f: S→T; 2. There is an onto function f: S→T; …

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).

From a total of 125 employees, 102 employees are assigned to perform ONLY ONE programming job, either system programming or application programming. And 10 employees are not assigned to perform any of those programming jobs. Overall, …

In a school, 100 students have access to three software packages A, B and C. 28 did not use any software 8 used only packages A 26 used only packages B 7 used only packages C 10 used all three packag…

1. Find these terms of the sequence {an}, where an = 2 · (−3)n + 5n. a) a0 b) a1 c) a4 d) a5

. What is the value of x after each of these statements is encountered in a computer program, if x = 1 before the statement is reached? a) if x + 2 = 3 then x := x + 1

For each of the following sets, a) S={1,2,3}, T ={a, b, c} b) S={a, b}, T ={1,2,3,4} c) S={1,2,3,4}, T ={a, b} Determine whether 1. There is a one-to-one function f:S→T; 2. There is an onto function f:S→T…

range of the function f(x) = 5x − 3 defined by the domain {0,1,2}

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…

Determine how many bit strings of length 5 can be formed, where the first bit is 0 and three consecutive 0s are not allowed.

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! …

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

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