Is the following notation for a product of all numbers of a set of real numbers strictly wrong?
$$\prod_{x \in X} x$$
I would argue that if the context shows that the multiplication is commutative, the notation is clear and a nice, short way to show what is wanted.
Context:
I used the notation in an exam and was deducted half the points (objective was writing down the log likelihood and I'm not a math but CS student). I think I read this notation in quite a few proofs and think the lecturer also used it a few times, but currently I'm failing to find any good examples.
Best Answer
This is a perfectly fine way of writing this product and it is used quite frequently - but definitely a little bit less frequently than the sum-symbol.
The only caveat (for both) is that it should be clear that this product makes sense: This is surely the case if $X$ is finite.
It will be a lot trickier if $X$ is not finite, as (just as with sums) the order of the factors is not defined, and it could also be the case that it diverges.
Some examples where these appear in "the wild":
https://en.wikipedia.org/wiki/Weierstrass_functions
https://en.wikipedia.org/wiki/Euler_product#Examples