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

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.

To understand how the modulo operator works in the given expressions, we need to understand what the symbol "≡3" represents. The symbol "≡3" denotes congruence modulo 3, which means that the numbers on both sides of the symbol have the same remainder when divided by 3.

Now let's break down the given expressions and understand how they can be negated using propositional notation.

(a) ∀x, (p(x) ∨ q(x))
In words, this statement can be read as "For every x in the set of integers, either x ≡3 1 or x ≡3 2."

(i) To express the negation using propositional notation, we can break it down into two parts:
- Negating "∀x" becomes "∃x," which means "There exists an x."
- Negating "(p(x) ∨ q(x))" becomes ¬(p(x) ∨ q(x)), which means "It is not the case that (p(x) ∨ q(x))."

So, the negation of the given statement is ∃x, ¬(p(x) ∨ q(x)).

(ii) In words, the negation of the given statement will be "There exists an x in the set of integers such that neither x ≡3 1 nor x ≡3 2."

(b) ∃x,∃y,p(x,y)
In words, this statement can be read as "There exist x and y in the set of real numbers such that x ≥ y."

(i) To express the negation using propositional notation, we negate the whole statement:
- Negating "∃x,∃y" becomes "∀x,∀y," which means "For every x and y."
- Negating "p(x,y)" becomes ¬p(x,y), which means "It is not the case that p(x,y)."

So, the negation of the given statement is ∀x,∀y,¬p(x,y).

(ii) In words, the negation of the given statement will be "For every x and y in the set of real numbers, it is not true that x ≥ y."

In summary, the modulo operator works by checking if two numbers have the same remainder when divided by a given divisor. In the given expressions, it is used to express congruence modulo 3, indicating that the numbers satisfy a certain condition related to remainders when divided by 3.