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Show that $X=\emptyset$ satisfies the following formula: "$\forall x\in X, x\subset X"$.

I´m not sure what this means. I think that it ask me to prove that $\emptyset={\{\emptyset\}}$. My question is if the last equation is true.

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No Its not true –  Amr Jan 5 '13 at 0:27
    
Can you show some element $x\in X$ which does not have that property ($x\subset X$)? –  Sigur Jan 5 '13 at 0:29
    
The last equation is not true, but it is not equivalent to the given question. The empty set has no members. (That is the definition of the empty set.) –  Qiaochu Yuan Jan 5 '13 at 0:30
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Any statement of the form "For every element of $\emptyset$, ..." is a vacuously true statement. –  user17762 Jan 5 '13 at 0:31
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Although $\emptyset$ and ${\{\emptyset\}}$ are different sets with different numbers of elements, they each satisfy "$\forall x\in X, x\subset X$". –  Henry Jan 5 '13 at 0:41

2 Answers 2

This isn't what it's asking you to prove; indeed, $\varnothing = \{ \varnothing \}$ is false.

Decode the statement: $(\forall x \in X)(x \subset X)$ is shorthand for $$\forall x(x \in X \to x \subset X)$$ But any statement of the form $\forall x(x \in \varnothing \to \psi)$ is vacuously true, for any $\psi$, since there is no $x$ for which $x \in \varnothing$.

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$\,\emptyset\,$ is a (the only) set with no elements at all, whereas the set $\,\{\emptyset\}\,$ has one element, namely $\,\emptyset\,$, and from here that they can't be the same.

Yet this is not what you're asked to prove, but rather that for any element $\,x\in\emptyset\,$ it is true that also $\,x\subset \emptyset\,$.

Perhaps you may want to google and read about a mathematical condition which is fulfilled in a vacuous or empty way, like saying: "I've no sister, but if I had one she'd be blonde"....well, kind of hard to discuss with that. Can you see the similarity?

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I don't think your example about sisters really conveys the idea of a vacuous truth; another example of the same form that is clearly informative is "you didn't drink the hemlock, but if you did, you'd be dead". –  Hurkyl Jan 5 '13 at 1:00
    
I think the hemlock example, the sister's and the basic idea of vacuously true statement are very close to each other. In my example, it could better be said, perhaps, "I've no sisters, but if I had one then Moscow would be Uruguay's Capital City", meaning: logically, from a non-checkable statement, or from a false one, you can deduce whatever you like...so for any element in the empty set, that element is pink, ugly and a subset of the empty set. –  DonAntonio Jan 5 '13 at 1:04
    
The problem is that's not how English works. The meaning of "if you did, you'd be dead" is more along the lines of "people who drink hemlock die", rather than the literal reading. This severely diminishes its value to those who don't yet understand logic, to the point of becoming very misleading. –  Hurkyl Jan 5 '13 at 1:30
    
Perhaps you're right and it's more of a language matter though, hopefully, not that confusing. –  DonAntonio Jan 5 '13 at 1:38

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