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I have sets $N = \{1, \ldots, n\}$ and $M = \{1, \ldots, m\}$.

When referring to a generic element of these sets, I typically use variables $i \in N$ and $j \in M$.

Is there any standard terminology to say up-front that, unless otherwise specified, when I write something like $x_i$ I implicitly mean $x_i \ (\forall i \in N$)?

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You should be wary of omitting quantifiers. It might save you writing time, but it's bad practice and can lead to ambiguous statements. For example, consider the statement:

There exists $j \in M$ such that $x_i \ne y_j$

With your convention, this could feasibly mean one of two things, namely:

  • For each $x_i$ there is some $y_j$ distinct from $x_i$
  • There is some $y_j$ which is distinct from all the $x_i$

In the case where $$N=\{1,2,3\},\ M=\{1,2\},\ x_1=1,\ x_2=2,\ x_3=3,\ y_1=1,\ y_2=2$$ the first of these statements is true and the second is false.

The only way to disambiguate between these two cases is to leave the quantifiers in the statement.

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  • $\begingroup$ That's a good point, and I agree with you in the general case. However, I am essentially defining a lot of values of the form $x_{i,j,k}$, and it is getting repetitive to keep saying $\forall i \in N, \forall j \in M, \forall k \in L$ after every such definition. In the cases I'm using it, I'm not ever combining $\exists$ and $\forall$, so the example you pointed out doesn't come up. $\endgroup$ – Eric Apr 9 '15 at 0:25
  • $\begingroup$ @Eric: In that case, if you're just defining terms, the context will probably speak for itself. $\endgroup$ – Clive Newstead Apr 9 '15 at 0:40

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