GATE CSE 2014 Set 2 | Question: 50

Question

Consider the following relation on subsets of the set $S$ of integers between $1$ and $2014$. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$.

Consider the following two statements:

• $S1$: There is a subset of $S$ that is larger than every other subset.
• $S2$: There is a subset of $S$ that is smaller than every other subset.

Which one of the following is $\text{CORRECT}$?

• A. Both $S1$ and $S2$ are true
• B. $S1$ is true and $S2$ is false
• C. $S2$ is true and $S1$ is false
• D. Neither $S1$ nor $S2$ is true

Comments

First take U={1,2,3,4} and V={1,2} (we can take any set other than ∅ and S)

why we cannot take  ∅ and s both are subset of s

---

For two distinct subsets U and V of S we say U<V if the minimum element in the symmetric difference of the two sets is in U :)

 

 

---

Very good question

Answer 1

The easiest wat to think of this question is assuming that

Largest subset is =  {2014}

Smallest subset is = {1}.

I thnk in this question empty subset won’t play any role.

Comments

if the minimum element in the symmetric difference of the two sets is in U  

and U and V are  two distinct subsets

think again

Answer 2

S1:   U=any subset of S other than phi

        V=phi 

So, yes there exists a subset i.e. phi larger than every other subset.

S2:  U={1,2,3,...2014} 

        V=any subset of S other than {1,2,3…...2014}

So, yes there exists a subset i.e. {1,2,3...2014} smaller than every other subset.

Answer 3

Consider a smaller set. Suppose S={1,2,3,4}

considering set S and ∅ (empty set)

First take U={1,2,3,4} and V={1,2} (we can take any set other than ∅ and S)

SD={3,4} (just exclude the elements which are common in the 2 sets)

Minimum element of SD is 3 which is in U and if we observe carefully minimum element will always be in U. Whatever the V is.

So acc. to the question {1,2,3,4} is smaller than any other subset of S.

Therefore, S2 is true.

Now consider U=∅ and V={1,2} (we can take any subset of S)

SD={1,2}

The symmetric difference will always be equal to V.

So minimum element of SD will always exist in V when U is ∅.

So acc. to the que, ∅ is greater than any other subset of S.

Therefore, S1 is also true.

This is true even when S={1,2,3,…,2014}.

So answer is A. Both S1 and S2 are true

Answer 4

.

Comments

nice and simple explanation , thanks akshay!