Quantifiers for multiple variables?

Question

I know about the universal quantifier(translated to "for all") $\forall$ and the existential quantifier(there exists) $\exists$. But I am not sure what the correct way is to use them for multiple multiple variables.

For example:

How would you use the quantifiers to express "for all x, y, c..."?
How would you use the quantifiers to express "for all x, y, c in $\mathbb{R}$..."?
How would you use the quantifiers to express "for all x, y, c in $\mathbb{R}$, there exists a, b such that..."?

$\forall x\forall y\forall c$, or $\forall x,y,c$ for short – J. W. Tanner Sep 08 '20 at 14:38 — J. W. Tanner, Sep 08 '20 at 14:38

score 2 · Answer 1 · answered Sep 08 '20 at 14:38

2

Both

$\forall x,y,c \dots$
$\forall x,y,c \in \mathbb R \dots$
$\forall x,y,c \in \mathbb R: \exists a, b: \dots$

and

$\forall x, \forall y, \forall c \dots$
$\forall x \in \mathbb R, \forall y \in \mathbb R, \forall c \in \mathbb R \dots$
$\forall x \in \mathbb R, \forall y \in \mathbb R, \forall c \in \mathbb R: \exists a, \exists b: \dots$

are acceptable.

answered Sep 08 '20 at 14:38

player3236

16,413

Ahh thanks! But why would you use a colon to separate the "for all" and "there exists" parts instead of a comma? Any specific reason? – That Guy Sep 08 '20 at 15:30
I didn't want $\mathbb R$ sticking to $\exists$. I have seen people use a comma ",", or bracket all the universal quantifier "phrases" (e.g. $(\forall x)(\exists y)$). It kinda means "such that". – player3236 Sep 08 '20 at 15:32
Just to point out that these colons/commas/periods inserted after $\forall x$ and $\exists x$ are unnecessary at best, and, frequently, gives rise to ambiguity, confusion, even misinterpretation. – ryang Mar 08 '23 at 05:34

Quantifiers for multiple variables?

1 Answers1