# What does this quantifier evaluates to?

I need help solving the following question, Suppose U=Z. Simplify the quantifer, and say that if it is true or false with the help of and example,

~$\exists$x $(x| x|5 \Rightarrow x|15)$

Note that |= modulo.

Now for all values it evaluates to false for me but I am not getting sure of my answer. Also I would like to mention that I just used x=0,x=1 to find out if the statement is true and false. Can there be any case that it evaluates to true?

## EDIT:

Thanks for the answers.
I tried solving it again and I have done it till, ~Ex(x|x|5->x|15)
Vx~(x|x|15->x|15)
As we know that ~(p->q)= ~pVq
therefore,
V(~(x|5))V(x|15)
Disproving by counter example,
I am out of ideas for the counter example, can anyone please give me a hint?

Thanks

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@KaratugOzanBircan: Yes it does mean that but why should I take x=5? – Fahad Uddin Nov 5 '11 at 11:59
Because 5|5 and 5|15 and thus 5 is a counterexample. – HbCwiRoJDp Nov 5 '11 at 12:06

## 1 Answer

I think you mean $\exists$ by E.

So, "~$\exists x : x | 5 \Rightarrow x|15$" means "There is no x such that if x divides 5, then it divides 15." Take x=5 and this is a counterexample. Hence the statement is false.

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you may use $\nexists$ notation – pedja Nov 5 '11 at 12:13