Solved: 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 you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis. f) Explain why these steps show that this formula is true whenever n is a positive integer. 5. Prove that 12+32+52+ + (2n + 1)² = (n + 1)

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…

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

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…

Determine whether the statement p→(q∨r) is equivalent to (p∧∼r)→q ?

Let A = {1, 2, 3, 4}. Define a relation R on A by a R b ⇐⇒ a + b ≤ 4 for every a, b ∈ A. (a) List all the elements of R. (b) Determine whether R has the following properties. If R has a certain property, prove this is so, otherwise, p…

Let A = {1,2,3,4}. Define a relation R on A by a R b ↔ a + b ≤ 4 for every a; b ϵ A. (a) List all the elements of R. (b) Determine whether R has the following properties. If R has a certain property, prove this is so, otherwise, provi…

1. Check whether the set S=R - {-1} 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. …

List of the members of these sets. (a){x|x is a real number such that x2 = 1.} (b){x|x is a positive integer less than 12} (c){x|x is the square of an integer and x < 100} (d){x|x is an integer such that x2 = 2}

For each of these pairs of sets,determine whether the ﬁrst is a subset of the second,the second is a subset of the ﬁrst, or neither is a subset of the other. (a) the set of airline ﬂights from New York to New Delhi, the set of nonstop…

For each of the following sets, determine whether 2 is an element of that set. (a){x∈R|x is an integer greater than 1} (b){x∈R|x is the square of an integer} (c){2 ,{2}} (d){{2},{{2}}} (e){{2},{2 ,{2}}} (f){{{2}}}

(i) Determine whether each of these statements is true or false. (a) 0∈∅ (b)∅∈{0} (c){0}⊂∅ (d)∅⊂{0} (e){0}∈{0} (f){0}⊂{0} (g){∅}⊆{∅} (ii) Determine whether these statements are true or false. (a)∅∈{∅} (b)∅∈{∅,{∅}} (c){∅}∈{∅} (d){∅}∈{{…

(a) Use a Venn diagram to illustrate the relationships A ⊆ B and B ⊆ C. (b) Use a Venn diagram to illustrate the relationships A ⊂ B and B ⊂ C. (c) Use a Venn diagram to illustrate the relationships A ⊂ B and A ⊂ C. (d) Suppose that A…

What is the cardinality of each of these sets? (a)∅ (b){∅} (c){∅,{∅}} (d){∅,{∅},{∅,{∅}}}

(i) Show that if A ⊆ C and B ⊆ D, then A×B ⊆ C ×D (ii) Let A ={a , b , c ,d}and B ={y , z}. Find (a) A×B (b) B×A

(i) Let A ={a , b , c}, B ={x , y}, and C ={0 , 1}. Find (a) A×B×C (b) C ×B×A (c) C ×A×B (d) B×B×B (ii) Let A ={1 , 2 , 3 , 4 , 5}and B ={0 , 3 , 6}. Find (a) A∪B (b) A∩B (c) A−B (d) B−A (iii) Let A ={a , b , c , d ,e}and B ={a , b …

Determine whether the following preposition is tautology, contradiction or contingency and explain the answer by your own words. (p↔q ) ⊕ ¬(q→p)

A set S is cardinally majorizable by a set T iff there exists a(n) ______________ from T to S.

Which of the following sets have the same cardinality? Select all that apply. LaTeX: \mathbb{N} N [0,1] LaTeX: \mathbb{R} R (0,1)

1. Let A and B be two non-empty finite sets. If cardinalities of the sets A, B, and are respectively 72, 28 and 13, then find the cardinality of the set .

need help with solving assignment

9. (i) Prove the identity laws in Table 1 by showing that (a) A ∪ ∅ = A (b) A ∩ U = A (ii) Prove the domination laws in Table 1 by showing that (a) A ∪ U = U (b) A ∩ ∅ = ∅ (iii) Prove the idempotant laws in Table 1 by showing that (a)…

10. Let A , B, and C be sets. Show that(a) (A ∪ B) ⊆ (A ∪ B ∪ C) (b) (A ∩ B ∩ C) ⊆ (A ∩ B) (c) (A − B) − C ⊆ (A − C) (d) (A − C) ∩ (C − B) = ∅ (e) (B − A) ∪ (C − B) = ∅

11. Show that if A and B are sets, then (a) A − B = A ∩ B (b) (A ∩ B) ∪ (A ∩ B) = A

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

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Question ID: mtid-5-stid-8-sqid-382-qpid-269