Is it valid to quantify two variables over the same universe using the same quantifier (like in $\forall a,b \in \mathbb{R} \:\: P(a,b)$)?

Question

I am wondering if we can quantify two variables over the same universe using the same quantifier, such as in $\forall a,b \in \mathbb{R}\:\: P(a,b).$

Are statements like this found in mathematical expressions? If so, do we regard this as quantifying a tuple of elements or two individual elements?

Answer 1

These are equivalent to one another

• $\forall a\,\forall b \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$
• $\forall a\,\forall b \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big)$
• $\forall (a,b) \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$
• $\forall (a,b) \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big),$

while these are interchangeable abbreviations of the above

• $\forall a{,}b {\in} \mathbb{R}\:\: P(a,b)$
• $\forall a{\in} \mathbb{R}\:\forall b {\in} \mathbb{R}\:\: P(a,b)$
• $\forall (a,b) {\in} \mathbb{R}^2\:\: P(a,b).$

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Addendum

The StackExchange Question

is there a way to prove that $$∀,\;(∀\:(≥⟹≥)⟹≥)\;?$$

suggests two other abbreviations:

• $\forall a,b \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$
• $\forall a,b \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big).$