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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

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

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

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

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

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

\forall k\in N(S(k)\rightarrow S(k+2)).

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

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 S_1, S_2 Always holds

(B) S_1 Always holds but S_2 may or may not hold.

(C) S_1 may or may not hold but S_2 Always holds.

(D) Both S_1, S_2 can never hold.

asked Aug 30, 2020 by gbeditor-2 (72,840 points)
edited Sep 9, 2020 by deepak-gatebook

1 Answer

0 votes
 
Best answer

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

\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:

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 deepak-gatebook (226,240 points)
Answer:

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

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

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
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