CMI2013-A-07

Question

Consider the following two statements.

1. There are infinitely many interesting whole numbers.
2. There are finitely many uninteresting whole numbers.

Which of the following is true?

• A. Statements $1$ and $2$ are equivalent.
• B. Statement $1$ implies statement $2$.
• C. Statement $2$ implies statement $1$.
• D. None of the above.

Comments

if one set goes -⚮ to +⚮

say it is interesting whole number

then uninteresting whole number contains nothing rt?

because both are whole numbers

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We can have two infinite subsets which are non intersecting from a given set- odd and even numbers, rational and irrational numbers etc are examples.

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yes it will be C) and we have to assume whole number range is infinite rt?

http://www.cmi.ac.in/admissions/sample-qp/pgcs2013-solutions.pdf

thank u @Manoj for this link

Answer 1

As we know that there are infinite whole numbers.

Now if we say that all even whole numbers are interesting whole number then we will have infinite interesting whole numbers and infinite uninteresting whole numbers (odd whole number ). 

Now if we say that  whole numbers between 1 to 100 are interesting whole number then we will have finite interesting whole numbers and infinite uninteresting whole numbers (whole numbers excluding 1 to 100 ).

So from second example we can say that -

If  there are finitely many uninteresting whole numbers, then there are infinitely many interesting whole numbers.
But we can't say anything about the converse i.e., if there are infinitely many interesting whole numbers then there are finitely many uninteresting whole numbers.

Ans- (C)

Comments

@srestha

even and odd is a well defined property of a number, isn't it ? we can say if a number is even, then number%2 will be 0. if remainder is 1 ,it is odd and even is the compliment of odd since there can be only two possible remainders when we divide with 2. 

here  question  doesn't give anything about whether they are compliment to each other or not. we are assuming just like even and odd, 'interesting and uninteresting' are compliment to each other? may be can be another class of whole numbers possible which are 'neither interesting nor uninteresting ' because we never know from question that only two classes exists. 

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Assumption is one of main thing in GATE like exam. Go all gate papers. You get many example like this

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 junk_mayavi they are complement of each other because of "un" in uninteresting or we can say not interesting

not is complement. 

Answer 2

Here is how I see this question being answered:

First, there are two facts to be understood:

1. The complement of a finite set of whole numbers is infinite:
   EXAMPLE:
   Let finite set be $\{x|x \in \mathbb{W} \text{ and } x \leq 10  \}$
   Then complement of this set is $\{x|x \in \mathbb{W} \text{ and } x > 10  \}$, which is infinite
    
2. The complement of a infinite set of whole numbers can be either finite or infinite:
   EXAMPLE:
   (i)  Complement of above infinite set is finite.
   (ii) If $S = \{x|x \in \mathbb{W} \text{ and  } x \text{  is odd}  \}$
        Then $\overline{S} = \{x|x \in \mathbb{W} \text{ and  } x \text{  is even}  \}$

Therefore we can say that Statement 2 implies Statement 1.
But the converse may or may not be true.

Therefore answer is (C)

Comments

Perfect Solution !!

Answer 3

Answer 4

According to me , stmt 1 implies stmt 2 and stmt 2 implies stmt 1 hence no conclusion followed so option D is correct.