Solved: 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 newspaper and the magazines, 5 people subscribed to magazines and were members of a CD club, and 10 people subscribed to the newspaper and were members of a mail CD club. Three people subscribed to both the newspaper and magazines and were members of a mail CD club. Make and fill in a Venn diagram to illustrate this situation

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

Given the following statements as premises: If he takes coffee, he does not drink milk. He eats crackers only if he drinks milk. He does not take soup unless he eats crackers. At noon today, he had coffee. Therefore he took soup at no…

Is the following argument valid? If Taxes are lowered, then income rise. Income rise. Therefore Taxes are lowered.

Write the following statement in symbolic form using quantifiers: I. All students have taken a course in Mathematics II. Some students are intelligent, but not hardworking.

Let A={1,2,3,4,5}, determine the truth value of the following: i.(∀x∈A)(x+3=10),ii.(∃x∈A)(x+3<5).

Write the negation of the flowing statement :∃x∈R,x>3⇒x^2>9.

Consider the two relationsdefined on the set of all people. (i) (a,b)  R, iff a is taller than b (ii) (a,b)  R, iff a and b were born on the same day. Determine whether the relations are reflexive, symmetric, antisymmetric and/or tr…

Let R be the relation {(1,1),(1,3),(2,2),(3,1),(3,2)}. (a) Find the 3x3 matrix MR representing R. (b) Find the matrix representing the transitive closure of R.

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. xRy if x>yx,y∈the set of all real numbers

Let R be the relation from A={2,3,4,5} to B={3,6,7,10} defined by ‘x divides y’, (a) thenR^(-1) is equal to: (b) Is R an equivalence relation?

For real number x and y, we write xRy⇔x-y+√2 is an irrational number. Is the relation (a) Equivalence (b) Partial order

Let R={(1,3),(4,2),(2,4),(2,3),(3,1) } be a relation on the A={1,2,3,4}. Find the transitive closure of R using Warshall’s algorithm.

Prove that the relation ‘’Superset of ’’ is a partial order relation on the power set of S.

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. m~n in Z if m=n mod 6. 51. Which of them are equivalence relations? (a) "less tha…

Draw the Hasse diagram for divisibility on the set {1,2,3,4,6,8,12}. Do the maximal, minimal elements exist? If so, what are they? What is the greatest element?

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. x~y in R if |x-y|<4

Let A={2,3,6,12}and let R and S be the following relations on A. xRyiff 2 / x–y, xSyiff 3 / x–y. Compute the following R^---,R^S,R U S,S^-1

Given the function F (X, Y , Z)=Σm(0,1, 2 , 4 , 6) answer the following questions: 1. Obtain the expression in the Canonical Disjunctive Normal Form 2. Obtain the expression in the Canonical Conjunctive Normal Form 3. Derive the t…

4. If the roots of the cubic az3 + bz2 + cz + d = 0 form an arithmetic progression α − β, α, α + β, prove that (2b 2 − 9ac)b + 27a 2d = 0.

Prove the following result by contradiction: Let f : X TO Y be a mapping. Suppose f (A INTERSECTION B) = f (A) INTERSECTION f (B) for all subsets A, B PROPER SET X , f (PHI) = PHI. Then f is a 1-1 mapping.

Solve the recurrence relation term an-4(term an-1)+5 term an-2- 2 term an-3=1+2^n

Solve the recurrence relation a_n-4a_n-1+5a_n-2-2a_n-3=1+2^n

Show that ( pimpliesq)^(qimplies~ p) equivalent~ p i) with truth table. ii) without truth table.

Give a self conjugate partition of 21, with justification.

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

Find out if the following functions are invertible or not, If it is invertible, then find the rule of the inverse (f^(-1) (x)) 1. f:k → k^+ f(x)=x^2 2. k^+ → k^+ f(x)=1/x 3. f:k^+ → k^+ f(x)=x^2

function f(x) = 5/9(x-32) converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion?

Formulate corresponding proof principles to prove the following properties about defined sets 1. A=B⇔A⊆B and B ⊆ A 2. De Morgan’s Law by mathematical induction 3. Laws for three non-empty finite sets A, B, and C

Let ρ ⊆ S × T, for finite sets S and T. Define fρ : S −→ P(T) be such that fρ(a) = {b | aρb}. a) Prove that ρ is reflexive if and only if a ∈ fρ(a), for every a ∈ S. b) Prove that ρ is symmetric if and only if a ∈ fρ(b), for every a ∈…

1/ Find weather the two function are invertible or not, if it is find out its inverse (f^1(x)) 1. f:[-π/2, π/2]→[-1,1]; f(x)=sin x 2. f:[0,π]→[-2,2]; f(x)=2cos x 2/ The function f(x)=5/9(x-32) converts Fahrenheit temperatures into C…

If A={2,4,6}and B={1,3,5}. Then \\(A\\cup B\\) is equal to__________ a.{1,2,3,4,6} b.{0,2,3,4,5,6} c.{1,2,4,5,6} d.{1,2,3,4,5,6}

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.

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

Produce truth tables for given Boolean expressions. 1) ĀB̅C+AB̅C̄+ABC+ĀBC̄ 2) (A+B+C)(A+B+C)(Ā+B+C̄)

Find the simplest form for the following boolean expressions using algebraic methods. 1. (A+B̄)(B+C)+(A+B)(C+Å̄) 2. (A+B)(AC+AC̄)+AB+B

Justify if the following operations on relevant sets are binary operations or not. 1) Multiplication and Division on se of Natural numbers 2) Subtraction and Addition on Set of Natural numbers 3) Exponential operation: (x,y)→x^y on Se…

Part 1 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 N…

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

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.

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