When is a set equal to empty set in contrary to being equal to a set containing only empty set as an element? I'm having trouble understanding the difference between following two statements:
$$\{X\subset \{a, b\} | \forall w \in X : |w| = 2\} = \{\emptyset \},$$
$$\{X \subset \{a, b\} | \exists w \in X : |w| = 2\} = \emptyset.$$
Why is one equal to a set having only empty set as an element, whereas the other one is equal to empty set itself? I mean neither one contains elements that fulfill the condition $|w| = 2$, so why aren't both just equal to $\emptyset$?
 A: This question in general seems to depend on what $a$ and $b$ are, for example if $a$ and $b$ both happen to have cardinality 2, then $\{X\subset \{a, b\} | \forall w \in X : |w| = 2\}$ is not empty (in that scenario it would be $\{\varnothing,\{a\},\{b\},\{a,b\}\}$.) But as you mentioned you know neither $a$ nor $b$ has cardinality 2, it's possible to give a definitive answer.

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*$Y=\{X\subset \{a, b\} | \forall w \in X : |w| = 2\}$ is a set of subsets of $\{a,b\}$. There are four subsets of $\{a,b\}$: $\varnothing$, $\{a\}$, $\{b\}$, and $\{a,b\}$. Since neither $a$ nor $b$ have cardinality 2, only one of these subsets is an $X$ where all members of $X$ have cardinality 2: $\varnothing$. For every member of $\varnothing$ (there are none), that member has cardinality 2. When said about the empty set, this is a vacuous truth, like the statement "I am undefeated in professional boxing". But for the other subsets $\{a\}$, $\{b\}$, and $\{a,b\}$, they all contain an element that fails to have cardinality 2, making the condition false. As a result, since the whole $Y=\{X\subset \{a, b\} | \forall w \in X : |w| = 2\}$ just contains the subsets where this condition is true, and $\varnothing$ is the only subset where this condition is true, $Y$ is the set containing just $\varnothing$, so $Y=\{\varnothing\}$.

*With $Z=\{X \subset \{a, b\} | \exists w \in X : |w| = 2\}$, this is again a set of subsets of $\{a,b\}$, but this time no set satisfies that property. There is no such thing as a vacuous truth for existence statements, if $X$ is empty then we know there certainly doesn't exist a $w\in X$ with any property whatsoever. None of the other subsets have any members with cardinality 2 either. So overall $Z=\{X \subset \{a, b\} | \exists w \in X : |w| = 2\}$ contains no subsets of $\{a,b\}$ (not even $\varnothing$), namely $Z$ is $\{\}$.

