I am given the following
universe of discourse: Z
predicate symbols: q
interpretation: $q(x,y) \implies x + y = 5$
I am told that $∃y.∀x. q(x, y)$ is false.
How is this possible? Isn't this indicating that:
"Every possible number added to another specific unique number equals 5"
How can this be false given that the universe of discourse is integers (thus negative numbers)? It may have have something to do with the order in which the universal quantifications are given.
The sentence $\exists y \: \forall x \: q(x,y)$ says "There exists a number $y$ such that if you take any number $x$, it will be true that $x + y = 5$.
Well, that's a false statement: there is no number $y$ that has that property. It is true that for every number $y$ there is some number $x$ such that $x + y = 5$, but given any number $y$, this is certainly not true for any number $x$. In fact, if it were true for every number $x$, then it should be true in particular for the number $y$, i.e. it would be true that $y + y = 5$. Clearly that is false given that we are dealing with integers.
Hence, the claim $\exists y \: \forall x \: q(x,y)$ is indeed false.
Your confusion is revealed by how you are reading the sentence. You say:
"Every possible number added to another specific unique number equals 5"
But (while trying to keep it as much the same as your sentence) it should be read as:
"Every possible other number added to a specific unique number equals 5"
When the quantifiers are in the order you give $y$ gets chosen first and has to work with every $x$. Your italicized text would be written $\forall x \exists y q(x,y)$ That statement is true because $x$ gets chosen first, then a $y$ needs to be found that makes $q(x,y)$ true.
The correct interpretation of $(\exists y)(\forall x) q(x, y)$ is "there exists a $y$ such that for all integers, $y$ plus that integer is equal to 5". You seem to be interpreting it as "for all $x$ there is a $y$ such that $x + y = 5$", which has the quantifiers backwards.