Set Addition vs. Set Union
Given two sets $A$ and $B$, what is the difference between $A + B$ and $A \cup B$? For example, if $A = \left\{ a, b, c \right\}$ and $B = \left\{ d, e, f \right\}$, what are $A + B$ and $A \cup B$, respectively?
Solution 1:
Usually $A+B$ is called the Minkowski sum and denotes the set of sums of an element of $A$ and an element of $B$. $A \cup B$ means the union with the common elements removed.
Example
$$\{1,2\} +\{1,2,4\} = \{ 1+1,1+2, 1+4 , 2+1, 2+2, 2+4 \} =\{2,3,4,5,6 \}$$ $$\{1,2\} \cup \{1,4,5\} = \{1,2,4,5 \}$$
Solution 2:
I have seen $A+B$ used in two contexts.
- When $A$ and $B$ are subsets of $\Bbb R$ (or generally something that $+$ is defined for) we sometimes write $A+B=\{a+b\mid a\in A\text{ and }b\in B\}$.
- Sometimes we write $A+B$ as a shorthand for the disjoint union of $A$ and $B$, which in the case above where the sets are disjoint is really just their union. Of course this is done where there is no danger of confusing with the previous definition (and often explicitly mentioned).