For each of the following cases, is ∀ k ∈ N ( P ( k...
For each of the following cases, is
true, false or dependent on the value of P(k)?
Answer & Explanation
First, notice that the sentence
is true whenever B is true. We use this fact in parts (a) and (c) below.
a) true (because for any natural number n, P(n) is true so P(2n) will also be true
Yes, the given statement (a) tells us that P(2k) is true regardless of quantification; thus, so is ; thus, (#) is true.
No. Here are two possibilities that are consistent with the given statement (b):
- statement (a) is true, in which case (#) is true,
- P(2) is false, in which case (#) is false.
Thus, we have insufficient information to conclude whether (#) is true or false.
Yes, the given statement (c) tells us that is true regardless of quantification; thus, (#) is true.