Notation and Quantifiers

Question

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.

Answer 1

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

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For some suggestions, you can see :

• Ethan Bloch, Proofs and Fundamentals : A First Course in Abstract Mathematics (2nd ed - 2011), Ch.2.6 Writing Mathematics, page 80-on,

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.