Solved: MATHEMATICAL INDUCTION AND RECURRENCE 1. What is the base case for inequality 3n > n2 , where n = 2? (2 pts) A. 3 > 1 B. 9 > 4 C. 6 > 4 D. 4 < 9 2.For the mathematical induction to be true, what type of number should be the value of n? A. natural number B. imaginary number C. rational number D. whole number 3. What would be the hypothesis of the mathematical induction for x(x + 1) < x! , where x ≥ 7? A. It is assumed that at x = k, k(k + 1)! < k! B. It is assumed that at x = k, k(k + 1)! > k! C. It is assumed that at x = k, k(k + 1)! < (k + 1)! D. It is assumed that at x = k, k(k + 1)(k + 2)! < k! 4. For any positive integer x, ________ is divisible by 5 (2 pts) A. 5x2 + 5 B. 2x + 4 C. x4 + 5x D. 3x2 + 2

. Let p and q be the propositions p: He is rich q: He is happy Write the following propositions using p and q and logical connectives. a. If he is rich, then he is unhappy. b. He is neither rich nor happy. c. It is necessary to be …

Give the power set of the following sets. (a) /0 (b) {1} (c) {1,2} (d) {1,2,3}

~(PÃ¢ q) Ã¢ p simplify by laws

state the value of x after the statement if P(x) then x = 1 is executed , when P(x) is the statement =x>1 , " if the value of x when this statement is reached is

Find, showing all working, a formula for the n-th term tn of the sequence (tn) defined by t1 = 5; tn = −7tn−1/3, n ≥ 2.

] Carry out the following calculations and conversions by hand (showing all your working). (a) Compute the sum (1011010)2 + (11011)2 (without changing the base). (b) Compute the sum (135)8 + (2357)8 (without changing the base). (c) Co…

Which of the following are propositions? Of those which aren’t, explain why. The baby is laughing. What’s that noise? 1 + 1 = 3 Get out of here! He ran in the race but slipped on a banana peel. What a glorious day it is! Santa …

Translate the following into symbols, given; p: It is hot. q: It is wet. r: I am tired. Example: 13. It is not the case that it is either hot or wet.  ~(p∨q) 1. It is not hot. 2. It’s hot and it’s wet. 3. It is not wet and I …

If 2 ≥ 3, then the cube of -1 is -1

1. Convert each of the following to their respective Decimal, Octal, Hexadecimal and binary representation: (a) (742)8 (b) (1011)2 (c) (47)10 (d) (3EAC)16

Let a and b be two Natural Numbers, such that the greatest common divisor of a and b is 63, and the least common multiple of a and b is 44452800. If ’b’ is an odd number, what is the minimum value of ’a’ possible? [Hint: a · b = gcd(a…

Solve the following: (a) 1231001 (mod 101) (b) 17123 (mod 13)

Let {an} be a sequence that satisfies the recurrence relation an=anÃ ¢ 1+3anÃ ¢ 2, for n=2,3,.,.,., where a0=1,a1=2. Find the values of a2,a3.

Find the characteristic root of the recurrence relation an=anÃ ¢ 1+2anÃ ¢ 2.

Consider the following argument: I will get grade A in this course or I will not graduate If I do not graduate, I will join the army I got grade A. Therefore, I will not join the army Is this a valid argument?

Prove that for any propositions p,q,r the compound proposition {pÃ ¢ (qÃ ¢ r)}Ã ¢ {(pÃ ¢ q)Ã ¢ (pÃ ¢ r)} is a tautology

Consider the following argument: I will get grade A in this course or I will not graduate If I do not graduate, I will join the army I got grade A. Therefore, I will not join the army Is this a valid argument?

Build a complete truth table and Show that they are Logically equivalent ¬(¬P ∧ Q) ∧ (P ∨Q)≡ p

For integers 𝑥 and 𝑦, consider the proposition ∃𝑥∀𝑦 (𝑥 + 𝑦 = 10). Is the proposition true, false, or does it depend on the values of 𝑥 and 𝑦?

In a class of n n students (we consider all students as distinct), we want to make k k groups where each group must contain at least 1 student. Let S(n,k) S(n,k) denote the number of ways in which such groups can be formed. Which o…

How many subsets of the set {1, 2, 3, 4, 5, 6, 7, 8} do not contain two consecutive integers ?

Establish the validity of the argument with the premises p -> (q -> r) , p \/ s , t ->q , ~s and ~r -> ~t

Prove that for all integer n3, P(n+1,3)−P(n,3)=3P(n,2).

Prove that n•P(n−1,n−1)= P(n,n).

find an increase in sequence of maximal links in a decrease in sequence of a maximal legs in a sequence 22,5,7,2,23,10, 15,21,3,17

Make a Fully truth table and Show that they are logically equivalent ⌐ (⌐p ∧ q) ∧ (p ∨q) ≡ p

Build a Fully truth table and identify if tautology, Contradiction or Contingency ¬(¬(P∧Q)↔(¬P)∨(¬Q))

Suppose that the domain of the predicate P (x) consists of 1, 2, 3, 4, and 5. Write out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

Given set A = {A, B, C, D, E, F, G, H} and B = {B, D, E, J, K}. Find for the: 1. A ∪ B 2. A ∩ B 3. A x B 4. B x A

Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that “Every positive integer is the sum of the squares of four integers.

The sum of the first n positive odd integers is n2. Establish the formula for the sum of the first n positive even integers and use proof by mathematical induction to prove its correctness.

prove that AUB = AU(B\A)

Let p and q be the propositions

M is a subset of the set of natural numbers. 10 elements of the set are prime numbers, and the rest are divisible by either 2, or 3, or 5. Determine the cardinality of the set if it contains: 70 numbers that are divisible by 2; 60 num…

x = 1 What is the value of x after each of these statements a. if x+2=3 then x:=x + 1 b. if (x+1=3) OR (2x+2=3) then x:=x+1 c. if (2x +3=5) AND (3x +4= 7) then x= x + 1 d. if (x+1=2) XOR (x+2=3) then x:=x+1 e. if x < 2 then x:=x…

If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area. * r→ q ↔ ¬p

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, −3, 2, −8, −3, −13, −8, −18, ? (c) 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, ? (2) Find a pair of numbers that provides a counte…

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, Ã ¢ 3, 2, Ã ¢ 8, Ã ¢ 3, Ã ¢ 13, Ã ¢ 8, Ã ¢ 18, ? (c) 1/2, 2/3, 3/4, 4/5…

(1) Use inductive reasoning to predict the next number in each list. (a) 1, 4, 9, 16, 25, 36, 49, ? (b) 2, 7, −3, 2, −8, −3, −13, −8, −18, ? (c) 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, ?

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a. 3x (x ² = 2) b. 3x (x ² = -1) c. Vx (x ²+2 Ã ¢ ¥ 1) d. Vx (x ² + x)

Let p and q be the propositions

Let p, q, and r be the propositions p : You have Covid. q : You miss the prelim examination. r : You pass the course. Express each of these propositions as an English sentence. a) p → q b) ¬q ↔ r c) q → ¬r d) p ∨ q ∨ r e) (p → …

What is discrete mathematocs and application

Make a Fully truth table and identify if it is a tautology, contradiction, contingency. ¬(¬(P∧Q)↔(¬P)∨(¬Q))

Use algebra of sets to prove that, [(𝐵 − 𝐴)' ∩ 𝐴] − 𝐴' = 𝐴

(4). Construct a difference table to predict the next term of each sequence. (a) 9, 4, 3, 12, 37, 84, ... (b) 10, 10, 12, 16, 22, 30, ... (7). Use Polya’s four-step problem-solving strategy and the problemsolving procedures to …

.Find these values. a) Ã ¢ 1.1Ã ¢ b) Ã ¢ 1.1Ã ¢ c) Ã ¢ Ã ¢ 0.1Ã ¢ d) Ã ¢ Ã ¢ 0.1Ã ¢ e) Ã ¢ 2.99Ã ¢ f ) Ã ¢ Ã ¢ 2.99Ã…

Let X={1,2} and Y={a,b,c}. List the elements in each set.

. Translate in two ways each of these statements into logi- cal expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. …

Prove that for all integer n>=3, P(n+1,3) - P(n,3) = 3P(n,2)

If ad-bc=0 then: a/c = b/d y = a/c = b/d So, y is a constant and is not one-to-one. Finding the inverse function: Let y = f(x) = (ax+b) / (cx+d) So y (cx+d) = ax+b So cxy + dy = ax+b So cxy - ax = b - dy So x (cy - a) = b …

Find the domain and range of the following function given by 𝑓(𝑥)=√(3𝑥−5)(𝑥+4)𝑥3−16𝑥.

Show that if 𝑎𝑑−𝑏𝑐 ≠ 0, then the function 𝑓(𝑥)=𝑎𝑥+𝑏/𝑐𝑥+𝑑 is one-to-one and find its inverse.

Prove that ( ) ( ) 1, 1 , nPnnPnn  − − = .

Show that ( ) ( ) ( ) 1, , 1 , CnkCnkCnk + = − + .

(a)How many bit strings of length 8 contain at least 6 ones? (b)How many bit strings of length 8 contain at least 3 ones and 3 zeros?

If n fair six - sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming 4 n  …

Let A={n∈N:20≤n<50} and B={n∈N:10

Consider the setsAandB,whereA={3,|B|}andB={1,|A|,|B|}.Whatare the sets?

(4) Construct a difference table to predict the next term of each sequence. (a) 9, 4, 3, 12, 37, 84, ... (b) 10, 10, 12, 16, 22, 30, ..

In a group of computer engineering students, some like algebra and others do not like algrebra. There is no student in this group who is indifferent or has no opinion. Two students are randomly selected from this group. (a) How many o…

Question#1 Use algebra of sets to prove the following: i. (𝐵 − 𝐴) ∪ (𝐶 − 𝐴) = (𝐵 ∪ 𝐶) − 𝐴 ii. [(𝐵 − 𝐴) c∩ 𝐴] − 𝐴c= 𝐴 iii. (𝐴𝑐 ∪ 𝐵)c ∩ Ac= ∅ Question#2 Use Mathematical induction to prove the following generalization of one of…

Find out reflexive, symmetric and transitive closure of following relation R. R = {(1,2), (2,3), (3,3)}

1. Write the multi-sets of prime factors of given numbers. a. 150 b. 450 c. 1250 2. Find the cardinalities of each multiset in parts 2-1. 3. Present the application offset and multiset in software engineering? Give specific progr…

1. Find the theta notation of the following: a. 6n ³+ 12n ² + 1, for nÃ ¢ ¥1 b. 3n ²+ 2n log2 n, for nÃ ¢ ¥1 2. Find the theta notation for the number of times the statement x:=x + 1 is executed. a. for i:=1 to 2n do…

Q#1 For any sets 𝐴, 𝐵 and 𝐶, if 𝐴 ⊆ 𝐵, then 𝐴 ∩ 𝐶 ⊆ 𝐵 ∩ 𝐶. Q#2 For any sets 𝐴, 𝐵 and 𝐶, (𝐴 × 𝐶) ∩ (𝐵 × 𝐷) = (𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷). Q#3 Given sets 𝐴, 𝐵 and 𝐶, prove that 𝐴 × (𝐵 ∩ 𝐶) = (𝐴 × 𝐵) ∩ (𝐴 × 𝐶) Q#4 Prove that If 𝐴 and 𝐵 are se…

Refer to the relation R on the set {1,2,3,4,5) defined by the rule (x, y) = R if 3 divides x - y 1. List the elements of R 2. List the elements of R-1 3. Find the domain of R 4. Find the range of R 5. Find the domain of…

Find all the minimal and maximal elements of A= {2 ,3 ,4 ,6 ,8 ,24 ,48} with partial order of divisibility using Hasse diagram

Use Polya’s four-step problem-solving strategy and the problem- solving procedures to solve each of the following exercises. (a) An android phone and a Bluetooth speaker together cost Php6500.00 . The phone costs Php2000.00 more …

Use deductive reasoning to show that the following procedure always produces a number that is equal to the original number. Procedure: Pick a number. Multiply the number by 6 and add 8. Divide the sum by 2, subtract twice the orig…

There are 4 adults and 6 children sitting around a round table. If there must be at least one child between any two adults, then how many ways are there for them to sit around the table? Rotations are considered the same, while re…

(a) How many bit strings of length 8 contain at least 6 ones? (b) How many bit strings of length 8 contain at least 3 ones and 3 zeros?

(a) Prove that for all integer n Ã ¢ ¥ 3, P (n +1,3) - P(n,3) = 3P(n,2) b)Prove that n.P(n-1,n-1) = P(n,n) c)show that C(n+1,k) = C (n,k-1) + C(n,k)

If n fairsix-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n ≥ 4?

Write an algorithm that outputs the smallest element in the sequence: SÃ ¢ , S2... Sn Use the procedure name find_small and output small. Make a recursive algorithm for finding the greatest common divisor of any non-negative i…

Let S be the following ordered sequence of elements S = {1, 2, 3, 4, 5, 6} and the universal set U be {1, 2, 3, 4, 5, 6}. Write down the characteristic vectors of • A = {1, 2, 4, 5}; • B = {3, 5}; • ∅; • A ∪ B; • A ∩ B; • A∪ ∼ …

Determine the number of subsets of size k of the set {1, 2, . . . , n} which do not contain consecutive integers. For instance, when n = 4 and k = 2, there are 3 such subsets, namely {1, 3}, {1, 4} and {2, 4}.

Determine the number of subsets of size k of the set {1, 2, . . . , n} which do not contain consecutive integers.

Determine the number of bijective functions f from {1, 2, . . . , 8} to itself such that f(i) 6= i for any even number i.

Find a system of distinct representatives for the following sets: {a, c, f}, {b, g}, {c, d, h}, {a, d, j}, {e, i, k}, {f, j, l}, {b, i, k}, {e, g}, {a, c, f, h, l}, {g, k}, {b, e, i}, {d, h, j}.

Find a system of distinct representatives for the following sets: {b,e,i,l}, {a,j,m}, {c,f,k}, {b,h,i,l}, {d,g,m}, {e,h,k,l}, {a,d,j}, {g,j,m}, {c,e,k}, {a,g,j}, {f,h,i}, {d,j,k,m}, {b,c,f}.

Fourteen chess players a, b, c, d, e, f, g, h, i, j, k, l, m, n are to be paired so they can play 7 games starting at the same time. The only pairs allowable are the following: {c,f}, {a,d}, {b,g}, {e,h}, {a,i}, {a,j}, {a,k}, {a,l}, {…

The rational numbers with 17 as a denominator are ... a) finite b) countably infinite c) uncountable d) non of these Choose 1 answer

Show that x2 is not O(x*log(x))

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

A particular algorithm increases in time as the number of operations, n, increases. Suppose the time complexity of this algorithm is given by: f(n) =4n3+5n2 * log(n). Show that f(n) is O(g(n)) for g(n) = n3.

Solve the congruence 5x≡1(mod12) Hint: 0≤x≤11

f(n)=4n 2 +5n 2 ∗log(n) , g(n) = n^3g(n)=n 3 Now to show f(n)=O(g(n))f(n)=O(g(n)) when number of operation increases i.e., n\rightarrow \inftyn→∞ then \frac{f(n)}{g(n)}=\frac{4n^3+5n^2.log\ n}{n^3}=4+…

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

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Question ID: mtid-5-stid-8-sqid-3062-qpid-1761