Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177

# DM - Grand Test -Q16

Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177
+1 vote

Suppose S(n) is a predicate on natural numbers (Positive Integers), n and suppose

$\dpi{100} \forall k\in N(S(k)\rightarrow S(k+2)).$

Now, if above assertion holds then Consider the following statements:

$\dpi{100} S_1:[\exists n S(2n)]\rightarrow \forall n S(2n+2)$

$\dpi{100} S_2:[\exists n S(n)]\rightarrow \forall n\exists m > n\,\,\, \,\,S(m)$

Which of the following is correct for above statements:

(A) Both Assertions $\dpi{100} S_1, S_2$ Always holds

(B) $\dpi{100} S_1$ Always holds but $\dpi{100} S_2$ may or may not hold.

(C) $\dpi{100} S_1$ may or may not hold but $\dpi{100} S_2$ Always holds.

(D) Both $\dpi{100} S_1, S_2$ can never hold.

edited Sep 9, 2020

Suppose S(n) is a predicate on natural numbers (Positive Integers), n and suppose

$\dpi{100} \forall k\in N(S(k)\rightarrow S(k+2)).$   \\ This says that If S(k) is true then S(k+2) is also true.

Now, if above assertion holds then Consider the following statements:

$\dpi{100} S_1:[\exists n S(2n)]\rightarrow \forall n S(2n+2)$

S1 is true when S(n) is true for all n. S1 is false when S(n) is true for all n > 9. So, S1 may or may not hold.

$\dpi{100} S_2:[\exists n S(n)]\rightarrow \forall n\exists m > n\,\,\, \,\,S(m)$

S2 is always true. If there exists some n for which S(n) is true then S(n+2) , S(n+4), S(n+6) and so on will also be true. So, If there exists some n for which S(n) is true then for infinite number of natural numbers S(n) will be true hence we can say that For all n there exists some m>n such that S(m) is true.

So, S2 always holds.

answered Sep 9, 2020 by (226,240 points)