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: Please prove the following without using outside lemmas,  "Let a, b ∈ Z. Prove that if ab is odd,…

A: To prove the statement "If ab is odd, then a^2 + b^2 is even" without using outside lemmas, we will…

Q: Let S be a set of n lines in the plane such that no two are parallel and no three meet in the same…

A: Let S be a set of n lines in a plane such that notwo are parallel and no three meet in the same…

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.

A:

Q: 4) Prove that SUT=SnT for all sets S and T

A: Given sets are S and T S¯∪T ¯ = S∩T

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

A:

Q: Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but…

A: Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but…

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

A: 1. option b

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: 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: Imagine a 3D plane P in your 3D scene. An infinite number of lines can lie on that plane. Consider…

A: We need to find the locus for the given scenario.

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: Let f(n) and g(n) be asymptotically positive functions. In the space provided, prove or disprove the…

A: The question has been answered in step2

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: Let f: A → B be a function with A₁, A₂ ≤ A. Determine if the following stateme true or false. Prove…

A: solutions in below step

Q: Let g: Z→ Z be defined by g(m) = m + 5. Is Is g a bijection? Explain. If g is a bijection, find its…

A: As per given information, solution is as follows:-

Q: 3. Let g: Z→ Z be defined by g(m) = m + 5. Is Is g a bijection? Explain. If g is a bijection, find…

A: If a function has inverse it must be bijective. Bijective Function: If a function is said as the…

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: Prove that the function y = n* + 3 can't belong or does not belong to the set of 0(1) using Big-0…

A: - We need to prove that the function y = n4 + 3 does not belong to the set of O(1).

Q: 3. Let TUV and S: V→ W. We have that SoT: U → W. Prove that (SoT)* = T* o S*. 4. Let v₁,.., Un EV be…

A: Question :  let T : U --&gt; V and S : V --&gt; W, we have that S o T : U --&gt; W, prove that (S o…

Q: By using the Big-O definition and formal proof, show and prove that a function y=n*+3 can't belong…

A: For a given function g(n), we have set O(g(n)) = { f(n) : there exist positive constants c and n0…

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: Given is a sequence a₁,02,...,an of numbers. We say that its subsequence aj. Aj Ajp wh

A: In computer science a sub-sequence is a kind of sequence created from the original sequence by…

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

A:

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

A:

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: 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: What is the asymptotic run time of the following function? Function f(n) r = 0 for i from 0 to n for…

A: The question has been answered in step2

Q: Let f : Z → Z be some function over the integers. Select an appro- priate proof technique (direct,…

A: Answer: I have given answered in the handwritten format in brief explanation

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.