Question

For each of the following examples:
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.