How would you write the following set? I am trying to get started on this question:
Let $A = \{a:a = 3^n, \ n \in \mathbb{Z}^+\}$
Let $B$ be defined as follows:


*

*$3 \in B$, and

*for any int $b$ and $c$, if $b \in B$ and $c \in B$, then $bc \in B$.


Then prove that $A$ is a subset of $B$, but that's not what I am worried about.
What does this set look like? My interpretation is $\{3, 9, 27, 81, 243, 2187, \dots\}$.
Does that seem correct? 
 A: B is not defined.  You require B to have a property.
Thus A subset B.  N also has that property.  B can
be any subset of N that includes A.  
Exercise.  Show A is the smallest set with that property.
A: $B$ has to include all of $A$.  $3$ is in it and if $b$ is in it so is $3b$ so all powers of $3$ are in it.  That means that all of $A$ is in it.  There is nothing in the definition of $B$ that says other things are not in it.  The positive or nonnegative integers satisfy the definition, as do the reals, etc.
A: You do not have enough definition to show entirely what $B$ may be, you only know two facts about some of its contents.   $3\in B$ and $\forall b\in\Bbb Z~\forall c\in\Bbb Z: ((b\in B\wedge c\in B)\to bc\in B)$.   
That is enough that you can use a proof by induction to demonstrate that $B$ will at least contain a certain subset of the integers ; perhaps other elements, but at least all of those.   What is this subset?


 Hint: for all positive natural $n$, $3^n$ is an integer (and also an element of $A$).

