For each of the following cases, is ∀ k ∈ N ( P ( k...

John Landry

John Landry

Answered

2022-07-15

For each of the following cases, is
k N ( P ( k ) P ( 2 k ) )
true, false or dependent on the value of P(k)?
a) n N P ( n ) ,
b) P ( 0 ) P ( 1 ) ,
c) n N P ( 2 n ) .

Answer & Explanation

Raul Garrett

Raul Garrett

Expert

2022-07-16Added 14 answers

Step 1
First, notice that the sentence
A B
is true whenever B is true. We use this fact in parts (a) and (c) below.
(#) k N ( P ( k ) P ( 2 k ) )
a) n N P ( n )
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 P ( k ) P ( 2 k ) ;; thus, (#) is true.
b) P ( 0 ) P ( 1 )
b) true
Step 2
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.
c) n N P ( 2 n )
c) true
Yes, the given statement (c) tells us that P ( k ) P ( 2 k ) is true regardless of quantification; thus, (#) is true.

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