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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 -Q15


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:(\forall n \leq 100(S(n)))\land (\forall n > 100 (\lnot S(n)))

S_2:S(1)\rightarrow \forall n S(2n + 1)

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 Never holds.

(C) S_1 Never holds but S_2 Always holds

(D) Both S_1, S_2 can hold but Not always.

asked Aug 30, 2020 by gbeditor-2 (72,840 points)
reshown Aug 31, 2020 by gbeditor-2

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:(\forall n \leq 100(S(n)))\land (\forall n > 100 (\lnot S(n)))

S1 is NEVER true. S1 becomeing true means S(n) is true for all n <= 100 and S(n) is false for all n > 100. But this is impossible because  If S(99) is true then S(101) will have to be true. So, S1 never holds.

S_2:S(1)\rightarrow \forall n S(2n + 1)

S2 is always true. If S(1) is true then S(3), S(5), S(7)... etc will be true so if S(1) is true then S(m) is true for all odd m. 

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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