Solved: a) Consider the whole of the English words set. Suppose an English word x is related to another English word y if x and y begin with the same letter. i) Show that this is an equivalence relation. ii) Compute C(quadratic) and C(rhombus) iii) How many equivalence classes are there in all, and why? iv) What is the partition of the English words under this relation? b) Consider Z, the set of integers. Suppose we define the relation: x is related to y if x - y > 3, x, y 0 Z. Determine whether or not the relation is i) reflexive ii) symmetric

Given the a Boolean function F(x,y,z)=(x+ y').z, write the sum-of-products expansion of F where all the variables x, y and z are used. Hint: Use some Huffington postulates, Boolean theorems and the unit property: p+p'=1

Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If it…

1. Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If…

Give partimos of R having (a) one block, (b) two blocks, (c) three blocks, (d) infinity many blocks

List all elements of the following sets as a set. All answers must be exact and not rounded 1.{q is an integer | q is a factor of 231} Describe the following sets using proper set-builder notation as explained in your book. You may n…

2. Produce truth tables for given Boolean expressions. i. A̅ B̅ C+A B̅ C̅ + ABC +A̅ B C̅ ii. (A+B̅+C)(A+B+C)(A̅+B+C̅)

Use the symbols ~, ^ and v, and write the following statements. both p or q and r

For the language L = {acnb | n > 0 }, which one of the following strings is not a part of the language ?

Part 2 1. Write the multisets of prime factors for the given numbers. I. 160 II. 120 III. 250 2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately. 3. Find the cardinalities of each multiset in pa…

1. Describe the characteristics of different binary operations that are performed on the same set. 2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on se of Natural …

Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way. 2. i. State the Lagrange’s theorem of group theory. ii. For a subgroup H of a group G…

1. Check whether the set is a group under the binary operation ‘*’defined as for any two elements . 2. i. State the relation between the order of a group and the number of binary operations that can be defined on that set. ii. How man…

1. Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If…

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra

Discuss two examples on binary trees both quantitatively and qualitatively.

2. Function converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion?

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights.

1. 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 practic…

If A= (a,b,c) then generate P(A) and 3 partition sets for A.

1. To access your Webwork account you are required to input the correct user- name and password. Suppose the you have auto ll for your username and are only required to input your password. You are granted \access" to your a…

Using the method of "searching for a falsifying truth assignment" prove the De Morgan's law

Let R be a relation on ℤ given by xRy if and only if x²-y² is divisible by 3. Show that this relation is an equivalence relation and find its corresponding equivalence classes.

‘A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the negative of B’. Is it true or false? give reasons.

Using Rules of Inference, can you show step by step that this argument is valid? NOT(IF p THEN q) AND p = NOT(q)

‘A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the negative of B’.

QUESTION 4 Below are a number of expressions. State which are terms, some are atomic wffs (well-formed formulae) and some are neither. a) Tet(y) b) Logician(john) c) father_of(quinn) d) 2 + y = z2 e) Angry(x; y;2:00)

QUESTION 2 Which of the following statements motivate the use of informal proof? Answer true or false to the following informal proof statements: a) Truth tables cannot demonstrate logical consequence for formulas containing more th…

QUESTION 3 Consider the arguments below and decide whether they are valid. If they are, write down an informal proof, phrased in complete, well-formed English sentences. If the argument is invalid, construct a counter example. In Qu…

QUESTION 5 In this question you have to construct formal proofs using the natural deduction rules. The Fitch system makes use of these rules. Remember that De Morgan’s laws and other tautologies are not permissible natural deduction…

4.2: For the proposition pairs below, create a truth table and compare each proposition’s truth profile: determine whether the pair is logically equivalent, contradictory, consistent or inconsistent. Example: (¬J ≡ K) with [(J → ¬K) …

Obtain the order of each element of S(℘) where S={1,2,3}

A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the negative of B’. is it true or false?

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example. (1) There are at least two ways of describing the set {7, 8....}. (2) Any function with domain R®R is a bi…

in a survey of 100 students, 56 wrote the Maths exams, 23 wrote psychology and 21 wrote the science exam. 12 wrote both maths and psychology exams, 9 write the maths and science exams and 6 wrote both psychology and science exams. 5 s…

When two sets A and B consist of the same elements, they are called_____sets a.equal b.compliment c.union d.difference

If kemi did not attend MTH 105 classes is (p), she will become angry (q). Kemi did not attend the class. Hence, she became angry. Write the statement symbolically.

If A=B, then A−B= _____________ a.B b.A\' c.A d.empty set

if kemi did not attend MTH 105 classes is (p), she will become angry (q). kemi did not attend the class, hence, she became angry. write the statement symbolically

find the sum \\ (\\bar{AB}+\\bar{BC}+\\bar{CD}+\\bar{DE}+\\bar{EF}\\)

Give four examples of implications and for each write down their contrapositive. Have two be real world examples and two be math examples.

1. Describe the characteristics of different binary operations that are performed on the same set. 2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on se of Natural …

Explain sexagesimal system used for operations on time. Give one example each of the problems involving subtraction and addition of time showing the error children make because of the sexagesimal system. How would you help children co…

Give two examples of an implication which is true, but whose converse is not true. One example should be a real-world example, while the other should be an example from math involving the integers.

Without changing their meanings , convert each of the following sentences into a sentence of the form “If P, then Q.” An integer is even provided it is not odd A geometric series with ratio r diverges whenever |r|>=1 Every polynomi…

Let P and Q be statements, then P↔Q is logically equivalent to

in a village, 180 houses have TV, 120 houses have radio. 80 houses have both. 125 houses have nothing. Find total number of houses

Reasoning missing number question.... Q- 4,3 and 7 have a triangular relation which get 42 as outcome and similarly 6,3 and 8 have same format relation with each other which get 91 as outcome so find the outcome if 5,3 and 9 have the …

A computer company receives 360 applications from computer graduates for a job planning a line of new Web servers. Suppose that 210 of these applicants majored in computer science, 147 majored in business and 51 majored in both subjec…

Translate the following statements into symbolic form using capital letters to represent affirmative Identify the main operator in the following propositions: ~[~(P & ~Q)] & ~(R –>~S) ~[~L v ~(J –> ~M)] ~[(H & ~L) & ~B] –> ~(~Y & ~K)…

Give four examples of implications and for each write down their contrapositive. Have two be real world examples and two be math examples.

1. Describe the characteristics of different binary operations that are performed on the same set. 2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on se of Natural …

Let I(x) be the statement “x has an Internet connection” and C(x,y) be the statement “x and y have chatted over the Internet,” where the domain for the variables x and y consists of all students in your class. Use quantiﬁers to expres…

Q.1) Proof the De Morgan’s Laws BY using truth table.

If kemi did not attend MTH 105 classes is (p), she will become angry (q). Kemi did not attend the class. Hence, she became angry. Write the statement symbolically.

identify the steps that are used to find the compound proposition for the statement "Either it is below freezing or it is snowing, but it is not snowing if it is below freezing."

How many ways are there to choose a valid password of the computer system

~(p • q) → (p ∨ r ) truth table

Prove Pigeon-Hole Principle: If a finite set S is partitioned into k sets, then at least one of the sets has |S|/k or more elements.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

Prove: If e is an edge in a simple closed path in G, then e belongs to some cycle.

In boolean algebra, proof: (a) x∨y = y if and only if x∧y = x

If x, y are real numbers such that ordered pairs (x + y, x -y) and (2x + 3y, 3x - 2y) are equal, then (x, y) is equal to

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

Let Q+ be the set of positive rational numbers. Prove that if x is in q+ there is some y in q+ such that y < x. Provide two proofs of this fact, one using a direct proof and one using a proof by contradiction.

a) Which of the following is a valid inference from the sentence, "c is a small dodecahedron but isn't either medium and left of b or right of d." 1. c is right of d 2. c is not a dodecahedron 3. c isn't either medium and l…

d) Assume ⊥ occurs in a subproof of the main proof. The appearance of ⊥ in this subproof indicates that: 1. The premises of the main proof are mutually inconsistent. 2. The assumption of the subproof is a TT-contradiction. 3. One of …

f) Suppose we are told that the following expression is true: P ↔ (Q ∧ ¬Q). What can we then conclude about P's truth? 1. P must be true 2. It is uncertain whether P is true or false 3. P must be false. g) Which of the following in…

h) Which assumptions are being referred to by the phrase "the assumptions in force" at a given step in a proof? 1. Only the premises at the top level of the proof. 2. Only the assumption of the subproof in which the step-in…

j) Suppose we translated Aristotle's famous sentence "All men are mortal" as ∀x(Man(x) ∧ Mortal(x)). Which domain could result in a counterexample to the claim that this is a good translation of the English sentence.? 1. A …

b) Suppose that b is small and d is large, and further that either b and d are identical or that c is a tetrahedron. Max believes that c must be a tetrahedron on the basis of this information. Which of the following is true of this si…

Let A and B be finite sets for which |A|=|B| and suppose f: A —> B. Prove that f is injective if and only if f is surjective

3. Stella Maris Schools, at one time, has 37 vacancies for teachers, out of which 22 are for English Language, 20 for History, 17 for Fine-Art. Of these vacancies, 11 are for both English Language and History, 8 for both, History and …

Chin-Tan found 52 action figures for his yard sale. He wants to put them in more than 1 box, but fewer than 5 boxes. Each box will have the same number of figures. How many boxes can Chin-Tan use? Explain.

Please read the following behavioral objective and identify the parts of the objectives as asked in the following questions.On the bus, Jonathan will sit quietly for 20 minutes, 80% of the time over 5 sessions.Choose the portion of th…

The fibonacci sequence is defined as x0 = 0, x1 = 1 and xn+2 = xn +xn+1 , for all non negative integers n prove that, xm = xr+1xm-r + xrxm-r for all integers m ≥ 1 and 0 ≤ r ≤ m-1 and xd divides xkd for all integers k and d

A car number plate consists of a letter, followed by five numbers, and ended with three letters. How many car number plates can be formed? if required there must not be the same letter and no equal lift, and how many car number plates…

Find in set builder notation the set of all positive integers a such that a≡137 (mod 5).

1. In a certain examination, 72 candidates offered Maths, 64 offered English, 62 offered French, 18 offered both Maths and English, 24 offered Maths and French, and 20 offered English and French. While 8 candidates offered all the thr…

In a road worthiness test on 40 cars. 60% passed. The number that failed had faults in clutch, brakes and steering as follows: Clutch only - 28; clutch and steering - 14; clutch, steering and brakes - 8; clutch and brakes - 20; brakes…

How can one prove a hamiltonian circuit?

Let A,B,C be subsets of a set. Prove that, A ∩ B ⊆ C iff A ⊆ Bcompliment ∪ C.

Does deductive reasoning always work? Elaborate your answer

A number of science students prepared for examination in physics, chemistry and mathematics. Out of the number 15 took physics,23 took Maths and 20 took chemistry. 6 took physics and maths, 9 took maths and chemistry. all those who to…

Develop truth tables and its corresponding Boolean equation for the following scenarios. a- The main function of automatic water level controller is to keep the water in the tank between two levels : low level indicated by sensor A an…

a) Given the propositions, and . Construct a Truth Table and a justifying a sentence to show these two propositions are logically equivalent. (8 marks) b) Construct a logic circuit for ¬ (((¬x ∧ ¬y) ∧ z) ∧ (x ˅ y)). (8 marks)

a) Provide an example of a relation T on set A = {1, c, 3} that is irreflexive and satisfies trichotomy. (What is Trichotomy) b) Let S= {(1, 2), (3, 4), (2, 3)} be a relation on set B = {1, 2, 3, 4}. Is S functional? Motivate your an…

Prove that n ! > 2^n for n a positive integer greater than or equal to 4. Prove that LHS = RHS

The binary operation on Z given by x mathrmast (y=1+xy) not in the question later appered when i pasted) is______ but not associative a.Associative b.Commutative c.Distributive d.Additive

1. Let A and B be two non-empty finite sets. If cardinalities of the sets A, B, and A∩B are respectively 72, 28 and 13, then find the cardinality of the set A∪B. 2. If n(A-B)=45, n(AUB )=110 and n(A∩B)=15, then find n(B). 3. If n(A)…

1. Write the multisets of prime factors for the given numbers. I. 160 II. 120 III. 250 2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately. 3. Find the cardinalities of each multiset in part 2-1.

1. Discuss two examples on binary trees both quantitatively and qualitatively

Write down the converse of each of the following statements: (2) i) If n =1 (mod 4) for a natural number n, then n = x^2+y^2 for two integers x and y.

Find the coefficient of x^18 in the expansion of (1-x-x^2)^10

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