Prove that for all integers a, b, and c, with a ̸= 0, if a|b and a|c, then a|(bx + cy)

Q: For each of the following pairs of functions ff and gg, circle one of the answers f \in o(g), f \in…

A: Ans: first of all you need to understand the basic concept of Notation: (1) 'o' Notation:  if…

Q: Prove or disprove that if a|bc, where a, b, and c are positive integers and a ≠ 0, then a|b or a|c.

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Q: Give a proof by cases that x ≤ |x| for all real numbers x.

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Q: f(n) such that f(n) ∈ O(n √ n) and f(n) ∈ Ω(n log n)) but f(n) ∈/ Θ(n √ n) and f(n) ∈/ Θ(n log n)).…

A: We know, big oh O, big Ω and big theta Θ notations are used to describe asymptotic upper bound,…

Q: Prove by Contradiction: For all integers x and y, x^2 - 4y is not equal to 2

A: Answer :

Q: 3. Using the pumping lemma and contradiction, prove that L = (a"b" n<m} is not regular.

A: We have to use the pumping lemma and contradiction theorem to prove the given statement: Statement:…

Q: Solve the following: (1) g(x)=tan(x³), find g'(x); (2) h(x)= (sin(3x))/2x, find h(x); (3) m(x) = In…

A: In this question we have the solve the given problem using MATLAB. Thou, the value y(x) is not…

Q: By using the definition of prove that 4√7n³ - 6n² + 5n − 3 € O(n¹.5). -

A: Defination of theta function Θ(g(n)) = { f(n): there exist positive constants c1, c2 and n0…

Q: Prove The range of πk − πk+1 is the same as the range of (πk − πk+1)(Ik − Ik+1). More specifically,…

A: solution:

Q: Prove, by finding constants C₁, C₂, and no that satisfy the definition of order of magnitude, that f…

A: Cv

Q: Given that h(p)=p2−3p+5, Find;h(x-3)

A: Given: h(p)=p2 + 3p + 5 then, by putting p=x-3 h(x-3) = (x-3)2 + 3(x-3) + 5           = x2 + 32…

Q: 1. Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer…

A: Answer: I have given answer in the handwritten format.

Q: To solve x such that 29x = 1 mod 100, we need to find the inverse of x. For such an inverse to…

A: If two integers a and b are coprimes, then only positive integer that is a divisor of both of them…

Q: 2. (c) Vs E R: s is irrational → s² is irrational.

A: Introduction

Q: Prove that (a + b) mod m = [(a mod m) + (b mod m)] mod m

A: The problem asks for a proof of a modular arithmetic property:Given: and (which means is a…

Q: Question Consider the language L = {a^ | n is a prime number}. Is this language regular or…

A: Given language is, L= {an | n is a prime number} Pumping Lemma for Regular languages states that, if…

Q: Asequencev(x,n)is bounded aboveif kER such that .A 2k V, EN .B X, Sk V, EN .C >k V EN

A: A sequence v( xn )is bounded above if k belongs to R such that ?

Q: 1. Big-O Notation Let fand g be functions from the set of integers or the set of real numbers to the…

A: Given that, f(x)= O(g(x)) that means f(x) is equal to big oh of g(x). Big-oh represents the tightest…

Q: S.O.P F(A, B,C, D) = E( 4 , 5 , 7 , 6 , 8 ,9 ) using P.O.S Karnaugh map.

A: Change the sign to ∏.The missing terms are 0, 1, 2, 3, 10, 11, 12, 13, 14, 15. Fill the K map as…

Q: Given functions f:R → R and g: R → R. Suppose that go f turns out as go f(x) = (x2 + 7]. Identify…

A: f:R→R f(x)=x2+2 g:R→R g(x)=x−1x​ fog(x)=f(g(x)) =f(x−1x​) =(x−1x​)2+2 =x2−2x+1x2​+2…

Q: Prove that I(X; Y |Z) ≥ I(X; Y ) . Note:  X,  Y, and Z are random variables.  X and Z are…

A: The question is asking to prove that the conditional mutual information I(X; Y |Z) is greater than…

Q: X = { ⟨Y⟩ | Y is a DFA and for every s in L(Y), ? = ab where a = b}. Show that X is decidable.

A: According to the question, we have to Express  problem as a language and show that it is decidable.…

Q: Prove or Disprove: For all real valued functions f, if f(n) is O(n²), then ƒ(2m) is 0(m²).

A: Since there is more than one question, I have provided answers for Q1 only. In step 2, I have…

Q: Let f(x) = x¹ Hx-x¹b, where H and b are constant, independent of x, and H is symmetric positive…

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Q: Propose and diagram an algorithm for the case that in P(x) for the case that includes negative…

A: Below i have answered:

Q: Prove or disprove. show your work. (a) for any integers n a and m: if both n and m are odd, then n-…

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Q: Prove that if x is odd, then x^2+ 3 is even.

A: Let x=3, Then x^2+ 3 = 3^ 2+3= 12 If x=5, Then  x^2+ 3 = 5^2+3 =28 If x=7, Then x^2+ 3 = 7^2+3=52…

Q: Prove that there are no integers u and v such that u is a prime greater than 5 and u=6v+3.

A: Step 1:- On Right Hand Side(RHS)  6v + 3= 3*(2v + 1) ........................taking 3 as a common

Q: Prove that Note: H(XY) ≤ H(X) using log sum inequality. -X and Y are random variables -X and Y are…

A: In information theory, entropy is a measure of uncertainty or randomness in a random variable.…

Q: Prove or Disprove: Vx, y ≤ Z, if x² (y² – 2y) is odd, then x and y are both odd. X

A:

Q: Prove that if n is a prime number, then √n is an irrational number.

A: Solution for given,  Prove that if n is a  prime number,  then root n is an irrational number.…

Q: Please do not give solution in image format thanku :Given two non-negative valued functions f(n)…

A: The question is asking whether it's possible to have two functions, let's call them f(n) and g(n),…

Q: Let f(n) = n2 and g(n) = 3n2 - 6n +4. Show that g(n) = 0(f(n)) by showing that there exist positive…

A:

Q: We want to prove that ƒ(n) = ²/1n² +8√n + 1 is O(n²) using the formal definition introduced in class…

A: Huffman coding is a no prefix code which is used to compress the data. In this coding technique, the…

Q: Assume that A is reducible to B. Show that if B is solvable in polynomial time, then A is also…

A: Answer is given below-

Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.