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$p$ : Every element in the empty set is greater than itself.

$\neg p$ : Some element in the empty set is smaller than or equals to itself.

I do not have the answers to this exercise, but it came out in my exams. I simply do not understand. There is nothing in an empty set! How can nothing be greater/equals/smaller/ to itself? If I had to choose an answer, it has to be $\neg p$ since nothing is equals to nothing...

What is the answer?

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3  
If $\neg p$ is true, then you'd have to find me an element of the empty set which is less than or equal to itself. Can you do that? –  Qiaochu Yuan Nov 15 '12 at 5:41
    
um.... nothing equals to nothing? Nevermind, I get the answer now! Thanks! –  Yellow Skies Nov 15 '12 at 5:44
3  
If you wanted to find me an element of the empty set which is less than or equal to itself, you'd first have to find me an element of the empty set... –  Qiaochu Yuan Nov 15 '12 at 5:45
    
Can I rewrite $p$ as "No element in the empty set is lesser or equals to itself"? Will that help? –  Yellow Skies Nov 15 '12 at 5:46
3  
See the Wikipedia page on vacuous truth. –  Bill Dubuque Nov 15 '12 at 6:05

2 Answers 2

up vote 4 down vote accepted

The negation of $\forall x \in S, x > x$ is $\exists x \in S, x \leq x$. If $S = \varnothing$, then there can't exist such an element in $S$, because there exists no element in $S$. Hence, $p$ is true. In general, every "$\forall$" property is true unless there exists a counter exemple.

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I see thanks! . –  Yellow Skies Nov 15 '12 at 5:44

Another approach based on the principle that anything follows from a falsehood or contradiction:

  1. $\forall a (\neg a\in\emptyset)$ (by definition)

  2. $x\in \emptyset$ (assume falsehood)

  3. $\neg P(x)$ (assume)

  4. $\neg x\in \emptyset$ (universal specification, 1)

  5. $x\in\emptyset \wedge\neg x\in\emptyset$ (2, 4)

  6. $\neg\neg P(x)$ (conclusion 3, 5)

  7. $P(x)$ (6)

  8. $\forall a (a\in \emptyset\rightarrow P(a))$ (conclusion 2, 7)

where $P$ is any unary predicate.

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Is there a reason that you use $\phi$ and not $\varnothing$? These are distinct symbols, and the empty set's symbol is not even derived from the Greek letter... –  Asaf Karagila Nov 26 '12 at 8:02
    
@AsafKaragila I didn't know there was an "emptyset" set symbol $\emptyset$ in Latex. Thanks. –  Dan Christensen Nov 26 '12 at 13:12

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