(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1 ≡ ¬(n = 1).
(b) Recall that a positive integer p is prime iff there do not exist a positive integers n and m, both greater than 1, such that p = nm. (I.e., Prime(p) means ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm).) Give a formal proof of the following: for any prime p, any positive integers n and m satisfying p = nm must satisfy either n = 1 or m = 1. Feel free to use the equivalence from part (a) as the rule “part (a)”.
a) Clearly, (n = 1) is True when n is 1. For any number other than 1, ¬(n = 1) is True. Hence, the statement that if n is a positive integer and n > 1, this implies that n is not equal to 1.
b)
Prime(p) ≡ ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm)
From part (a), n > 1 ≡ ¬(n = 1) and m > 1 ≡ ¬(m = 1), we get
Prime(p) ≡ ¬∃n ∃m (¬(n = 1) ∧ ¬(m = 1) ∧ p = nm)
Prime(p) ≡ ∀n ∀m (¬(¬(n = 1) ∧ ¬(m = 1) ∧ p = nm))
Prime(p) ≡ ∀n ∀m ((n = 1) v (m = 1) v p = nm) (proved)
(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1...
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