For each of the following examples:

Question

i) Express the negation of the statement using propositional notation, simplifying as much as possible,
and
ii) Express the statement (and its negation) in words, for the provided propositional functions.
(a) ∀x, (p(x) ∨ q(x)), where x ∈ Z, p(x) is the statement x ≡3 1, and q(x) is the statement x ≡3 2.
(b) ∃x,∃y,p(x,y), where x,y ∈ R, and p(x,y) is the statement x ≥ y.

I'm confused how the mod works in to this expression

Answer 1

statement: for all x, either p(x) or q(x)

There is some x where p(x) and q(x) are both false.

The mod works just like any other function

x ≡3 1
Since x is congruent to 3 (mod 1) makes no sense, you must mean
x ≡ 1 (mod 3)
We all know what that means, so just negating the congruency is enough.
Negating both p and q just means that 3 divides x.