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Let $N$ and $X$ be arbitrary sets, and let $f:N\times X\to\mathbb{R}$ be a function. Let this function satisfy $f(i,x)=f(j,x)$ for all pairs of elements $i,j\in N$ and for every element $x\in X$. I want to write this formally, but I’m unsure what’s the best way of doing so.

Option 1: $(\forall i,j\in N)(\forall x\in X)(f(i,x)=f(j,x))$

Option 2: $((\forall i,j\in N)(\forall x\in X))(f(i,x)=f(j,x))$

Option 3: $(\forall i,j\in N)\wedge(\forall x\in X)(f(i,x)=f(j,x))$

Option 4: $((\forall i,j\in N)\wedge(\forall x\in X))(f(i,x)=f(j,x))$

Options 1 & 2 are identical, but option 2 encloses the expression $(\forall i,j\in N)(\forall x\in X)$ between parentheses, while option 1 does not. Similarly, options 3 & 4 are identical, but option 4 encloses the expression $(\forall i,j\in N)\wedge(\forall x\in X)$ between parentheses, while option 3 does not. The difference between options 1 & 2 vs. 2 & 3 is the use of $\wedge$ to indicate “and”.

I think that my favourite option is $4$, but I have no solid argument in favour of it (or against the other options). Could anyone help me choose the best notation?

EoDmnFOr3q
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1 Answers1

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Options 3 and 4 are clunky at best and actually incorrect at worst; the use of "and" to combine quantifiers like that is not how mathematical "and" works (it only operates on propositions, which can have a truth value, which quantifiers don't).

Between options 1 and 2, option 1 is less clunky because it has fewer parentheses. I would just write it in natural language the way you did in your second sentence to avoid clunkiness, but if forced to use symbols I would personally write

$$\forall i, j \in N, x \in X : f(i, x) = f(j, x)$$

specifically to minimize the number of parentheses.

Qiaochu Yuan
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