Solved: Discuss how you can efficiently use Graph Theory to construct a route planning of a project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as millage of the vehicle, etc.) as much as you can. Essentially consider the two fold, - Routes with shortest distance(Quick route travelling by own vehicle) - Route with the lowest cost

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

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Prove that the relation R:{(x,y)|x-y is divisible by 3} is an Equivalence relation.(R is defined over Z) L2 CO2 10

(a) By the definition of piece arrow- P\downarrow Q=P↓Q= ~(P\lor Q)(P∨Q) P\downarrow QP↓Q =~(P\lor P)(P∨P) We have derived that P\downarrow PP↓P is logically equivalent with ~P ~P=P&b…

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…

F. 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

F. 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?

Given the following 2 premises, 1. 𝑝 → (𝑞 ∨ 𝑟) 2. 𝑞 → 𝑠 Prove 𝑝 → (𝑟 ∨ 𝑠) is valid using the Proof by Contradiction method.

Use the Principle of Mathematical Induction to prove that (2𝑛)! /(2n)𝑛! is odd for all positive integers.

(P^Q^R)V (¬P^Q^R)V(¬P^¬Q^¬R)

Use the truth tables method to determine whether (¬p ∨ q) ∧ (q → ¬r ∧ ¬p) ∧ (p ∧ r) is satisfiable.

Suppose that A,B and C are sets such that A is the improper subset of B and B is the improper subset of C

(p^q)v(P ^ Q ^ R)

Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor antisy…

Justify whether the given operations on relevant sets are binary operations or not. - Multiplication and Division on set of natural numbers - Subtraction and Addition on Set of natural numbers - Exponential operation:…

Use the Principle of Mathematical Induction to prove that ((2n) !)/(2^n n !)((2n)!)/(2 n n!) is odd for all positive integers.

How to draw a network diagram in mathematical foundations of computer science (10mark)

1. Write the multisets (bags) of prime factors of given numbers. i. 160 ii. 120 iii. 250 2. Write the multiplicities of each element of multisets (bags) in Part 2-1(i,ii,iii) separately. 3. Determine the…

4. Let P(n) be the statement that 1+2+...+n'3D (n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do…

show that factorial function is promitive recursive

An algorithm is a _________ set of precise instructions for performing computation.

Given the following 2 premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) is valid using the Proof by Contradiction method.

Draw the Hasse diagrams of all partial ordered sets with at most 4 elements. Which of these are lattices?

Determine whether each of these functions is a bijection from R to R (real).a) f (x) = 2x + 1b) f (x) = x2 + 1c) f (x) = x3

Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

show that 12+32+52+.....(2n+1)2=(n+1)(2n+1)(2n+3)/3 where n is a nonnegative integer.

A Function f : Z→Z, f(x) = x+5, is invertible since it has the inverse function g : Z→Z, g(x) = x−5.

Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cr…

An electrician charges a base fee of $70 plus $50 for each hour of work. Create a table that shows the amount the electrician charges for 1,2,3, and 4 hours of work. Let x represent the number of hours and y represent the amount charg…

Given The Following 2 Premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) Is Valid Using The Proof By Contradiction Method. [Hint: Use A Combination Of Equivalence Laws And Rules Of Inference To Solve This Question]

Construct the network who's activity and the relationship Activity:A,D,E can start simultaneously Activity:B,C>A;G,F>D,C;H>E,F

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Construct the network who's activity and the relationship Activity:A,C,D can start simultaneously Activity:E>B,C;F,G>D;H,I>E,F;J>I,G;K>H;B>A

Construct the network who's activity and the relationship Activity:A

Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive

Find the domain and range of this functions a) The function that assigns to each pair of positive integer the maximum of these two integers b) The function that assigns to each positive integer the number of the digits 0,1,2,3,4,5,6…

which of the following statements are true? justify your answers i) the contrapositive of "not a ⇒ not b' is "a&b', where a and b are two statements. ii) any set can be represented by the listing method

Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither …

is 6x+1 < (2x+4) + (4x-2) proposition?

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What is the image (range) of the function that assigns the square of an integer to this integer

Construct the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-00…

prove a --> ( b V c ) using contradiction method and combination of inference rules and equivalence laws from these premises : 1. a --> ( d V b ) 2. d --> c

Use rules of inference to show that the hypothesis "All lions are fierce","Some Lions do not drink coffee" imply the conclusion "Some fierce creatures do not drink coffee".

Suppose you want to encode messages containing only the following characters with their given respective frequencies: B: 55 D: 15 E: 80 G: 5 U: 45 (a) What is the minimum length bit string required to encode each character with a dis…

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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)

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an + an-1 - 10an-2 + 8an-3

{F} 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?

{F} 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

{F} 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 arrow…

Solution: injective function Definition: A function f: A → B is said to be a one - one function or injective mapping if different elements of A have different f images in B. A function f is injective if and only if whenever f(x) = f…

{F} 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

{F} Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

{F} Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor an…

Determine wheter 4 is the element of each of the following sets A: P={1,2,3,4,5} B: R={prime numbers}

Given P(x,y) :y=x+5 ,determine the truth value of each of the following if the universe is the set of real numbers.

Identify the mapping diagram that represents the relation and determine whether the relation is a function(−3,−6),(−1,−6),(5,−6),(8,−6)

A survey of 100 students produced the following information: 30 speak English, 45 speak Hindi and 60 speak Marathi. It is found that 10 speak both English and Hindi, 20 speak both Hindi and Marathi, 15 speak both English and Marathi a…

4. A survey of 100 students produced the following information: 30 speak English, 45 speak Hindi and 60 speak Marathi. It is found that 10 speak both. English and Hindi, 20 speak both Hindi and Marathi, 15 speak both English and Marat…

In a survey of 130 people, the following data were collected: 106 people subscribed to the newspaper, 29 people subscribed to magazines, and 17 people were members of a mail CD club. Seventeen subscribed to both the newspa…

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1. There are 41 candidates approved to be allotted as the F.D.R.E. council of cabinets. Historic facts of the candidates are shown below, 8 were served as former cabinet members,14 were university presidents, 15 served as members …

Solve the following set of recurrence relations and initial conditions3an-1+n^2-3,n>1,a0=1

What is mean by logic

Let R br a realtion defined from the set A={1,2,3,4} to the set B={2,3,4,5}as a€A, b€B aRb <-> a+b=5 1. What are the orderd pairs in the relation R 2. Represent R with a matrix

There are three kinds of people on an island: knights who always tell the truth, knaves who always lie, and spies who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight,…

an=3an−1+n2−3,n≥,a0=1

Show that if any five numbers from 1 to 8 are chosen, then two of them will add up to 9.

if R={(1,2),(2,1),(3,1),(2,3)} be a relation defined on A={1,2,3)then transitive closure of R is

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find A ∩ B ∩ C. A ∪ B ∪ C. (A ∪ B) ∩ C. (A ∩ B) ∪ C.

Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B ∪ C by showing each side is a subset of the other side. using a membership table.

Draw the Venn diagrams for each of these combinations of the sets A, B, and C. A ∩ (B − C) (A ∩ B) ∪ (A ∩ C) (A ∩ ) ∪ (A ∩ )

If A={1,2,3} , find the relation R={(a, b) ∈ A²|a>b}

Suppose that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express each of these sets with bit strings where the ith bit in the string is 1 if i is in the set and 0 otherwise. {3, 4, 5}=0011100000 {1, 3, 6, 10} = 101…

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