Q:

Is in general true that $[a,b] + [c,d] = [a+c,b+d]$?

Let $a,b,c,d\in \mathbb{R}$ with $a\leq b $ and $c\leq d$. If we consider the Minkowski sum $A+ B = \{x+y : x\in A,y\in B \},$ is true in general that $[a,b] + [c,d] = [a+c, b+d] $?

A:

The set $X = [a, b] + [c, d]$ clearly lies within the closed interval $[a + c, b + d]$ and contains its endpoints. Since $X$ is the image of the connected set $[a, b]\times [c, d]$ under the continuous map $(x, y) \to x + y$, it is itself connected, and thus must be exactly $[a+c, b + d]$.

