Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
This is solved by a standard method in additive combinatorics:
Construct an injective map $A \times S_- \to S_+ \times S_+$.
This goes as follows: First associate to each element $d \in S_-$ a pair of elements $(b,c) \in A^2$ such that $b-c=d$.
Of course this is not necessarily unique but we can just choose any fixed representative.
Then we can map a pair $(a,d)=(a,b-c) \in A \times S_-$ naturally to $(a+b,a+c) \in S_+ \times S_+$.
This is certainly a well-defined map but we need to make sure that it is injective i.e. we need to reconstruct $a$ and $d$ only knowing $a+b$ and $a+c$.
Of course we have $d=b-c=(a+b)-(a+c)$ so we can certainly reconstruct $d$. But then we also know $b$ and $c$ since we chose fixed representatives in the beginning. So we know $b$ and $a+b$ and hence also $a$ and our map is indeed injective.
Actually this is just a special case of the well-knwon triangle inequality of Ruzsa which states that
\[\vert B\vert \cdot \vert A-C\vert \le \vert A-B\vert \cdot \vert B-C\vert.\]Here, we can take $C=A, B=-A$.