Definition of A Specific Type

Question

I was studying David Marker's Model Theory book and saw this notion for the first time in page 117: $$tp^N( \overline{a} /A)$$
I think the author has not given it a definition before. Could you say it's defiinition and why it is a complete type!

Answer 1

$\overline{a}$ is a finite tuple from $N$, and $A$ is some set of parameters. $tp^N(\overline{a}/A)$ is just the type - in $N$ - of the tuple $\overline{a}$ over the set $A$. That is, the collection of all formulas-with-parameters $\varphi(\overline{x}, \overline{c})$, with $\overline{c}\in A$, such that $N\models \varphi(\overline{a},\overline{c})$.

It's complete because for every such $\varphi(\overline{x},\overline{c})$, either $N\models \varphi(\overline{a},\overline{c})$ or $N\models\neg \varphi(\overline{a},\overline{c})$, so either $\varphi(\overline{x},\overline{c})\in tp^N(\overline{a}/A)$ or $\neg\varphi(\overline{x},\overline{c})\in tp^N(\overline{a}/A)$.