Create a mathematical proof to prove the following:
Given an integer n, and a list of integers such that the
numbers in the list sum up to n. Prove that the product of a list
of numbers is maximized when all the numbers in that list are 3's,
except for one of the numbers being either a 2 or 4, depending on
the remainder of n when divided by 3.
Consider the following functions from ℤ × ℤ → ℤ. Which
functions are onto? Justify your answer by proving the function is
onto or providing a counterexample and explaining why it is a
counterexample.
(a) f(x,y) = xy + 3
(b) f(x,y) = | xy | + 10
(c) f(x,y) = ⌊( x+y ) /
3⌋
Consider the following functions from ℤ × ℤ → ℤ. Which
functions are onto? Justify your answer by proving the function is
onto or providing a counterexample and explaining why it is a
counterexample.
(a) f(x,y) = xy + 3
(b) f(x,y) = | xy | + 10
(c) f(x,y) = ⌊( x+y ) /
3⌋
For any Gaussian Integer z ∈ ℤ[i] with z = a+bi , define N(z)
=a2 + b2. Using the division algorithm for
the Gaussian Integers, we have show that there is at least one pair
of Gaussian integers q and r such that w = qz + r with N(r) <
N(z).
(a) Assuming z does not divide w, show that there are always two
such pairs.
(b) Fine Gaussian integers z and w such that there are four
pairs...
1. Which of the following predicate calculus statements is
true?
Question 1 options:
∀n ∈ ℤ, n + 1 > n
∃n ∈ ℤ, n + 1 < n
∀n ∈ ℤ, n > 2n
∀n ∈ ℤ, 2n > n
2. Which of the following is the correct predicate calculus
translation of the sentence "Some natural numbers are at least
100"?
Question 2 options:
∃n ∈ ℕ, n > 100
∀n ∈ ℕ, n ≥ 100
∃n ∈ ℕ,...
In this exercise we outline a proof of the following statement,
which we will be taking for granted in our proof of the division
theorem: If a, b ∈ Z with b > 0, the set
S = {a − bq : q ∈ Z and a − bq ≥ 0}
has a least element.
(a) Prove the claim in the case 0 ∈ S.
(b) Prove the claim in the case 0 ∈/ S and a > 0. (0...