Question

For each set of conditions below, give an example of a predicate P(n) defined on N that satisfy those conditions (and justify your example), or explain why such a predicate cannot exist.

(a) P(n) is True for n ≤ 5 and n = 8; False for all other natural numbers.

(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.

(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is False.

(d) P(1) is True, P(k) ⇒ P(k + 1) is False for all k ∈N.

Answer 1

a) is true for only and false for all other naturals

b)

Then is true but P(1) is false

c) so P(1) and P(2) are true but is false

d)

The P(1) is true but is false (one is odd means the other is even)

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