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25. Let Q : N → {true, false} be a predicate with domain the natural numbers N = {0, 1, 2, . . .} and defined by the statement Q(k) = “k is a prime number”. Identify the following propositions as true or false. Unless otherwise stated, only use. • the fact that the only even prime is 2, and

25. Let Q : N → {true, false} be a predicate with domain the natural numbers N = {0, 1, 2, . . .} and defined by the statement Q(k) = “k is a prime number”. Identify the following propositions as true or false. Unless otherwise stated, only use. • the fact that the only even prime is 2, and

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 49E: Show that if the statement is assumed to be true for , then it can be proved to be true for . Is...
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25. Let Q : N → {true, false} be a predicate with domain the natural numbers N =
{0, 1, 2, . . .} and defined by the statement Q(k) = “k is a prime number”. Identify the
following propositions as true or false. Unless otherwise stated, only use.
• the fact that the only even prime is 2, and
• basic arithmetic with natural numbers.


Show all you working and reasoning.
(a) Q(1), Q(2), Q(3), Q(4)
(b) For all k ∈ N, k > 2, we have ¬Q(k) ∨ ¬Q(k + 3).
(c) ∃k ∈ N : Q(2k)
(d) There exists a k ∈ N such that for all ℓ > k we have that Q(ℓ) is false. You may
use external sources to prove that the proposition is true or false. 

Can you please help me to show working out for the question 25? Thank you

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