Sum of sets

In $\mathbb{R}$ we define the sum of sets as follows:

$$A+B= \{a+b, \; a \in A, \; b \in B \}$$

Find the sum $[a, b]+[c, d]$. 

Solution

 
We give  a geometrical proof based on the following image.

Based on the definition we gave , it is sufficient to add to the one of the two sets the elements of the one set and join the images. In the image above this is achieved by taking the image of $[a, b]$ through all functions $f(x)=m+x , \; m \in [c, d]$ and joining their images. The result is the projection of the parallelogramm ${\rm K}\Lambda {\rm M}{\rm N}$ onto the $y$ axis.