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 ) .

Name: 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 ) .
Uploaded: 2022-02-23T17:22:57+00:00
Description: 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 ) .

Question

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  )  .

Raul Garrett · Accepted Answer

Step 1First, notice that the sentence  A      →    Bis 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 trueYes, 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) trueStep 2No. 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) trueYes, the given statement (c) tells us that   P  (  k  )          ⟹        P  (  2  k  ) is true regardless of quantification; thus, (#) is true.