Terminology for when a variable is implicitly a member of some set?

Question

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 $x_i \ (\forall i \in N$)?

Answer 1

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.