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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?

  • 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?2012-11-15
  • 0
    um.... nothing equals to nothing? Nevermind, I get the answer now! Thanks!2012-11-15
  • 4
    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...2012-11-15
  • 0
    Can I rewrite $p$ as "No element in the empty set is lesser or equals to itself"? Will that help?2012-11-15
  • 0
    Does it help you?2012-11-15
  • 0
    I guess so... There is no element in the empty set, and hence, the statement must be true regardless.2012-11-15
  • 4
    See the Wikipedia page on [vacuous truth.](http://en.wikipedia.org/wiki/Vacuous_truth)2012-11-15

2 Answers 2

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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! .2012-11-15
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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.

  • 1
    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...2012-11-26
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    @AsafKaragila I didn't know there was an "emptyset" set symbol $\emptyset$ in Latex. Thanks.2012-11-26
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    I didn't know there was a $\varnothing$ in Latex either.2015-04-29