Union between sets: notation

Question

I read in a not recent book, whose name I cannot remember, that instead of using the notation $A\cup B$ to denote the union of sets, the notation $A+B$ was used. Now is $A+B$ always used? If yes I wished to know what is the name of such a sum and what is the current definition with some very simple examples that I can show to my 14-year-old students (first year science high school).

Answer 1

Set theoretically,

$$A \cup B:=\{x: \text{$x \in A$ or $x \in B$}\},$$

I.e., the set of points that are in $A$ or $B$ or both.

While

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

I.e., the set of sums of elements from $A$ together with elements of $B$.

Hope this helps!!

For example, if $\Bbb{I}$ denotes the irrationals, then

$$\Bbb{I} \cup \Bbb{Q}=\Bbb{R}$$

But

$$\Bbb{I}+\Bbb{Q}=\{a+b: a \in \Bbb{I}, b \in \Bbb{Q}\}.$$

Which is not all of $\Bbb{R}$. Namely, $a+b \in \Bbb{R} \setminus \Bbb{Q}$ where $a \in \Bbb{R} \setminus \Bbb{Q}, b \in \Bbb{Q}$. This is a known fact which requires some proof of course.

Also a note, or remark on notation, $A-B$ could means elements in $A$ but not in $B$ or set of differences ($\{a-b:a \in A,b \in B\}$), which is why i write set of elements in $A$ but not in $B$ as $A \setminus B$. Please upvote if you agree with everything I've said! :) cheers to more maths!