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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?

  • 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.

  • 0
    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.

  • 0
    I didn't know there was a $\varnothing$ in Latex either.2015-04-29