How do I translate this pattern of sentence into a quantified statement using logical operators? (It's basically introducing a new symbol, "b" in this case, in a proof.)
So a = xb where b is the integer yc.
Do I go
$∃b ∈ Z, (b = yc ∧ a = xb)$
or
$(∃b ∈ Z ∧ b = yc), a = xb$
or
$∃b = yc ∈ Z, a = xb$
or neither?
What if defining "b" takes even more statements? Thanks in advance.
Are you asking which is the clearest to an audience? In that case I'd personally go with $$ \exists b \in \mathbb{Z},(b=y c \wedge a=x b) $$ because it's a simple existence statement with the conditions. The second and third are a little more sloppy, for instance the third you may as well say $yc \in \mathbb{Z}$ and define $b = yc$.
I hope I understood and answered your question properly!