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I was wondering what is a natural way to write certain formal expressions, without make them look too cumbersome.

In particular, what I learned from various books is that, when we deal with the existential quantifier, we use the symbol "$:$" and then "$\wedge$" or simply a comma ",". Thus, for example, we have

  • $ \exists x: P(x) \wedge Q(x)$,
  • or $ \exists x: P(x), Q(x)$.

In particular, the last expression looks kinda better, because it uses the quantifier (and quantifiers are not yet considered too formal), but it does not use "$\wedge$", which should already look too logical.

Now, how do we write expressions with "$\forall$"?

In general, for what I have studied (e.g. Velleman's "How to prove it"), we should write something like $$ \forall x \ ( \ P(x) \Rightarrow Q(x) \ ).$$

However, I do have the feeling that this is already considered a bit too cumbersome (I am referring in particular to the brackets), if – for example – we are using it to specify something about $x$ in a definition.

Thus, is –for example – $$ \forall x, \ \ P(x) \Rightarrow Q(x) \ $$ wrong?

I think so, but I don't know another option to write the same.

As always, any feedback is greatly appreciated.
Thank you for your time.

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1 Answer 1

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There are various notations for restricted (or bounded) quantifiers.

You can use also :

$(\forall x)_P Q(x)$

which is the abbreviation for : $\forall x(P(x) \Rightarrow Q(x))$,

and :

$(\exists x)_P Q(x)$

which abbreviates : $\exists x (P(x) \land Q(x))$.


For some suggestions, you can see :

and specifically the sub-sections [page 86] :

2.6.9. Distinguish Formal vs. Informal Writing

2.6.10. Miscellaneous Writing Tips.

Regarding quantifiers, see Ch.1.5 Quantifiers, page 34-on.

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  • $\begingroup$ Thanks a lot. However, my question does not concern other formal way of writing expressions with quantifiers. It is rather about how to write informal expressions still using – partly – quantifiers. $\endgroup$
    – Kolmin
    Commented Feb 12, 2015 at 15:58
  • $\begingroup$ Again, thanks a lot for the references. However, now a very straightforward question arise. If you have an expression – not in the body of the text! (such as \begin{equation}, etc) – which of the two expressions I will write down do you choose? $\endgroup$
    – Kolmin
    Commented Feb 12, 2015 at 16:29
  • $\begingroup$ 1. $X := \{ x \ | \ \forall X \in \wp (X) \ ( x \in X \ ) \ \}$ $\endgroup$
    – Kolmin
    Commented Feb 12, 2015 at 16:30
  • $\begingroup$ 2. $X := \{ x \ | \ \forall X \in \wp (X), \ x \in X \ \ \}$ $\endgroup$
    – Kolmin
    Commented Feb 12, 2015 at 16:32
  • $\begingroup$ To me, it seems the second is just wrong, but maybe the first is considered too cumbersome. $\endgroup$
    – Kolmin
    Commented Feb 12, 2015 at 16:33

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