Solved: 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 0,1,2,3,4,5,6,7,8,9 that do not appear as decimal digits of the integer.

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.

There are four train routes between A and B, and three train routes between B and C. Find the number of ways that a person can travel by train: i. from A to C by way of B; ii. roundtrip from A to C by way of B; iii. roundtrip from A t…

Combinations: (a) A class contains 10 students with 6 men and 4 women. Find the number of ways to: i. Select a 4-member committee from the students ii. Select a 4-member committee with 2 men and 2 women iii. Elect a president, vice pr…

Using contrapositive method, prove the following. (i) For any integer n, if n 2 − 6n + 5 is even then n is odd. (ii) For any integers a and b, if a + b is even then a and b are even. (iii) For any integer n, if n 2 is odd then n is od…

Using direct method, prove the following. (i) If n is an even integer then -n is even. (ii) If n is an even integer then 3n + 5 is odd. (iii) If n is an odd integer then n 2 + 3n is even. (iv) If m is an even integer and n is an odd i…

What is the De Morgan’s law for quantifiers? (b) Write the negation of following statements. (i) ∃x (x 2 + 2 < 1) (ii) ∀x (x - 2 ≥ 3) (iii) ∀x (x 2 ≥ 0 and x + 2 < 1) (iv) ∃x (x 2 + 2 < 0 or x - 5 ≥ 0) (v) ∀x (x - 2 ≥ 3 and x 3 + 5 ≤ …

(a) Let P(x) be the statement x 2 ≥ x. (i) What are truth values of the propositions P(1) , P(-1) , P(0) and P( 1 2 )? (ii) What is the truth value of the proposition ∀xP(x), where the domain consists of all real numbers? (iii) What i…

Let n be a natural number. Use mathematical induction to prove that 4 n−1 > n2 for all n ≥ 3.

Which of the following pairs of statements are logically equivalent? I. Statement 1: It is not true that either I watched 'Crashlanding on You' or you watched 'Money Heist'. Statement 2: I did not watch 'Crashlanding on You' and you w…

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…

2. Let P(x) and Q(x) be the statements x3 < 50 and x+2 < 5 respectively. Find the truth values of following quantifications, where the domain consists of all real numbers. (i) ∃x P(x) (ii) ∀x P(x) (iii) ∀x Q(x) (iv) ∃x Q(x) (v) ∀x (¬P…

2. Obtain the product of sums canonical form of the following formula i)(P^Q^R)V (¬P^Q^R)V(¬P^¬Q^¬R)

Construct a proof for the five color theorem for every planar graph.

2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on set of Natural numbers ii. Subtraction and Addition on Set of Natural numbers iii. Exponential operation: on Set …

1. (i) Prove that if m and n are integers and mn is even, then m is even or n is even. (ii) Show that if n is an integer and n3 + 5 is odd, then n is even using (a) a proof by contraposition (b) a proof by contradiction (iii) Prove th…

2. (i) Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even. (ii) Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd. (iii) Prove that m2 = n2 if and only if m = n or m = -…

Show that these statements about the integer x are equivalent. (i) 3x + 2 is even. (ii) x + 5 is odd. (iii) x2 is even.

Let P(n) be the statement that 12 + 22 + ... + n2 = n 6(n + 1)(2n + 1) for 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? …

5. (i) Prove that 12 + 32 + 52 + ... + (2n + 1)2 = (n+1) 3 (2n + 1)(2n + 3), whenever n is a nonnegative integer. (ii) Prove that 3 + 3.5 + 3.52 + ... + 3.5n = 3(5n+1 - 1) / 4, whenever n is a nonnegative integer.

6. (i) Show that postage of 24 cents or more can be achieved by using only 5-cent and 7-cent stamps. (ii) c1 = 0 and cn = cn/2 + n2 for all n > 1. (a) Compute c2,c3,c4 and c5. (b) Prove that cn < 4n2 for all n ≥ 1.

If p→q is false can you find the truth value of ~ (p ∧ q) →q? Explain your answer

Part (a): Determine whether the statement p→(q∨r) is equivalent to (p∧∼r)→q ? Part (b): A survey team in Food Street is surveying of liking and disliking of people there. The food items to be considered in survey are of three types, …

Part (a): Let Aand B are any sets then show that A-(A∩B)=(A∩A^c)∪(A∩B^c) by using membership table. Part (b): Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions. A∩B≠ϕ,B∩C≠ϕ,A∩C=ϕ,A⊈B,C⊈B. Part (c): Find…

Let A= {1, 2, 3, 4, 5}then define a relation R on A as (a.b)∈R iff a≤b and Relation Ton A as (a,b)∈T iff a/b. Represent R by a matrix. Is R Reflexive? Transitive? Give a valid reason for your answer. Is T Antisymmetric? Give a valid …

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2} 1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain. 2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One 3. D…

Let A = {1, 2, 3, 4, 5} then define a relation R on A as (a.b)∈R iff a≤b and Relation Ton A as (a,b)∈T iff a/b. Represent R by an matrix. Is R Reflexive? Transitive? Give a valid reason for your answer. Is T Antisymmetric? Give a val…

Part (a): Let A and B are any sets then show that A-(A∩B)=(A∩A^c)∪(A∩B^c) by using membership table. Part (b): Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions. A∩B≠ϕ,B∩C≠ϕ,A∩C=ϕ,A⊈B,C⊈B.

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2} 1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain. 2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One 3. D…

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 0,1,2,3,4,5,6,7,8,9 that do not appear as decimal digits of the integer.

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