How would you write the following set?

Question

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:

1. $3 \in B$, and
2. 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?

Answer 1

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?

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

Answer 2

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.

Answer 3

$B$ 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.