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.
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.
(e) For all k1, k2 ∈ N, k1, k2 > 2, we have that (Q(k1) ∧ Q(k2)) =⇒ (¬Q(k1 · k2) ∧ ¬Q(k1 · k2 + 1))
(f) There exists k1, k2 ∈ N, k1, k2 > 1, k1 ̸= k2, such that (Q(k1)∧Q(k2)∧Q(k1·k2+1)).
(g) For all n ∈ N, there exists k1, k2, k3, . . . , km ∈ N such that Q(ki) is true for all 1 ≤ i ≤ m, and n = k1 · k2 · . . . · km.
Can you please help me with this discrete question part e) f) g)?
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