negation

[topaz kestrel]

When finding the negation of a statement, I am confused by the phrase "such that". I cannot find the proper position for it. Can someone please tell me how can I find the right place for the phrase?

[radiant tundra]

I'm not sure if this is what others do, but I tend to have "such that" be implicit after every ∃

So, in this case, I'd write

¬∀a ∈ ℤ, (∃ b ∈ ℕ, {such that} 3a+b≠5)
≡
∃a ∈ ℤ, {such that} (∀b ∈ ℕ, 3a+b=5)

I'm not sure if this is the norm though

I find that saying it outloud helps too.

For every element x in the set of natural numbers such that P(x), ... Is equivalent to ∀x ∈ {n ∈ ℕ | P(n)}, ..., And not ∀ x ∈ ℕ, P(x). So, in this case, it wouldn't make sense for there to be a "such that" after the quantifier bit

[topaz kestrel]

Thank you so much

[native marsh]

@topaz kestrelIf you want to, you can treat "such that" as an extra condition you are setting for a certain variable