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:
$S_1$: There is a subset of $S$ that is larger than every other subset.
$S_2$: There is a subset of $S$ that is smaller than every other subset.
Which one of the following is CORRECT?
A) Both $S_1$ and $S_2$ are true
B) $S_1$ is true and $S_2$ is false
C) $S_2$ is true and $S_1$ is false
D) Neither $S_1$ nor $S_2$ is true
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Answer given-
$S_1$ is true because NULL set is smaller than every other set.
$S_2$ is true because the UNIVERSAL set $\{1, 2, …, 2014\}$ is larger than every other set.
But i think its correct explanation is this-
$S_1$ is true because $∅$ is greater than any other subset of $S$.
$S_2$ is true because $\{1, 2, 3, …., 2014\}$ is a subset of $S$ that is smaller than every other subset.
Please tell which is correct and why other is wrong?
Any help is appreciated in advance!