# Notation and Quantifiers

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.

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.

• 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. Commented Feb 12, 2015 at 15:58
• 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 , etc) – which of the two expressions I will write down do you choose? Commented Feb 12, 2015 at 16:29
• 1. $X := \{ x \ | \ \forall X \in \wp (X) \ ( x \in X \ ) \ \}$ Commented Feb 12, 2015 at 16:30
• 2. $X := \{ x \ | \ \forall X \in \wp (X), \ x \in X \ \ \}$ Commented Feb 12, 2015 at 16:32
• To me, it seems the second is just wrong, but maybe the first is considered too cumbersome. Commented Feb 12, 2015 at 16:33