subset with a special property

Question

I was reading a book and there was a theorem about all the subsets of $\mathbb{R}$ with the following property: $a+b\not \in A \Leftarrow a,b \in A$.

I tried to think of a simple subset with the following property, but my creativity has failed me. I am looking for the simplest subset.

Edit: I didn't understand the property right. The set $\{1\}$ is a good example.

Edit: The other thing that I didn't understand is how to prove that there is an maximum set out of all those subsets of $\mathbb{R}$ with the mentioned property.

Answer 1

How about $A=\{1\}$?

Or, for that matter, $A=\varnothing$, in which case your (backwards) implication is vacuously true.

Answer 2

Henning already provided the simplest examples. Here is a more complicated one:

Let $A$ be a $\mathbb Q$-basis for the vector space $\mathbb R$. Then, for all $a, b \in A$ we have $a+b \not \in A$ -- by the $\mathbb Q$-linear independence of that set. Moreover, $A$ has the same cardinality as $\mathbb R$.