Existential Notation: Is "$\exists mn$" the same as "$\exists m\exists n$"?

Question

I'm doing an exercise for Discrete Mathematics, chapter Logic, topic Quantifiers. However, I never seen this kind of notation: "$\exists mn$".

Is "$\exists mn$" same as "$\exists m \exists n$"?

Answer 1

Is "$\exists mn$" same as "$\exists m \exists n$"?

Yes, exactly. This is just a shorthand / abbreviation. If you want to be completely formal about it, you would write out $\exists m \; \exists n$.

Answer 2

Yes, this is because $\exists m, \exists n, P(m,n)$ is true if and only if $\exists n, \exists m, P(m,n).$ So you can think of $\{m,n\}$ as a block variable where order does not matter and instead write $\exists \{m,n\}, P(m,n)$ or informally $\exists m,n, P(m,n).$