How To Analyze Statements of Quantifiers

Question

Having trouble figuring out how to interpret Universal Quantifiers, from my book there's two sets of statements. Assuming x,y and z are real numbers, determine the truth value of each statement

(a): $\forall x \text{ and } \forall y,\exists z \ni y-z=x$

(b): $\forall x \text{ and } \forall y,\exists z \ni xz=y$

According to the book, a is true and b is false but I'm having trouble figuring out why (along with how to analyze existential and universal qualifiers.

I'm reading statement a as "for any given value of $x$ and any given value of $y$, there exists a value $z$ (which isn't fixed but can be different depending on the values of $x$ and $y$) Rewriting the equation yields $z=x-y$. Logically taking the difference between any possible values of $x$ and $y$ should produce a real number $z$ thus statement a is true.

For statement b if you rearrange the equation you get $z=\frac{y}{x}$, $z$ will be a real number for all values except for $x=0$. If $x=0,0*z=y$ then this statement is false because, $0*z=y$ if and only if $y=0$. Thus the statement is false because the equation is not true for all possible values of $x$ and all possible values of $y$.

Does this mean revising statement b to read

(b): $\forall x>0 \text{ and } \forall y,\exists z \ni xz=y$

Would make statement b a true statement?

Is this the proper method for analyzing quantifiers? Analyzing quantifiers seems less straightforward than truth tables (where you can simply construct a table for statements like $p \Rightarrow q$) so I'm not sure if I'm doing it properly.

Answer 1

Your analysis is correct. Reading them as you have is the way to understand them. I much prefer reading arguments that use the words rather than the logical symbols since that saves me the translation - and presumably saves the writer the translation too, since we tend to think with words.

Truth tables are in fact easier, though I prefer words even when there are no quantifiers.