First order logic

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February 10th 2011, 06:32 AM
Mike12

First order logic

can any one give me a first-order sentence that says there exist exactly three elements. Please explain it to me.
Thanks
February 10th 2011, 06:37 AM
Ackbeet

How about this (assuming you mean that there exist exactly three distinct elements):

$(\exists x)(\exists y)(\exists z)\left[\neg(x=y)\land\neg(x=z)\land\neg(y=z)\land(\forall t)((t=x)\lor(t=y)\lor(t=z))\right].$

This says that there are three elements that are mutually unequal, and if you consider any element at all, it must be equal to one of those three elements. Does that do the trick?
February 10th 2011, 07:03 AM
Mike12

Thank you very much about that. I found this in a book but without example about the first order sentence that says there are three elements.
So , as I understood , I just write a sentence that has three elements which are distinct. is that correct
Thanks
February 10th 2011, 07:04 AM
Mike12

Thanks
February 10th 2011, 07:06 AM
Ackbeet

Quote:

Originally Posted by Mike12

Thank you very much about that. I found this in a book but without example about the first order sentence that says there are three elements.
So , as I understood , I just write a sentence that has three elements which are distinct. is that correct
Thanks

Well, if you're trying to say that there are exactly three elements, then you have to have a way to say that there aren't any more than three. That's what I'm doing with the t in my expression above. So, the first three AND'ed clauses ensure that you have at least three distinct elements. The t business ensures that everything is one of the three elements you've already constructed, thus guaranteeing that there aren't any more than three elements.
February 10th 2011, 07:10 AM
FernandoRevilla

Quote:

Originally Posted by Mike12

Thanks

Don't waste a message. Better press the Thanks button :)

Fernando Revilla