# Logic behind addition of two intervals

I would like to know the logic behind adding two intervals $$[a, b]$$ $$[c, d]$$ together and for the sake of simplicity let's say that the numbers in the interval are just integers and not real numbers. I know that the idea is to just add the lowest and highest of the two $$[a+c, b+d]$$ and this is the result of the addition.

My question is whether this is just defined to be like this mathematically or is there a real world example that can shed some light on why it is calculated so?

• This is simply a consequence of adding linear inequalities. Mar 13, 2021 at 10:30
• Suppose we know that a < x < b, and c < x < d. Then (a + c) < x < (b + d). Thus, ([a, b] + [c, d]) = [(a + c), (b + d)]. Mar 13, 2021 at 10:31
• @Doug, that shows $[a,b]+[c,d]$ is contained in $[a+c,b+d]$, but you need something more to show they're equal. Mar 13, 2021 at 10:52
• @GerryMyerson You're right. Oops. Mar 13, 2021 at 11:03
• @DougSpoonwood - Did you mean "$a \le x \le b$ and $c \le y \le d$. Then $(a+c) \le x + y \le (b+d)$"? Without $x + y$, what you wrote has far more wrong than Gerry Myerson pointed out. (And of course $[\;,\;]$ mean $\le$, not $<$.) Mar 13, 2021 at 17:58

Here's a proof that if $$r$$ is in $$[a+c,b+d]$$ then $$r=x+y$$ for some $$x$$ in $$[a,b]$$ and some $$y$$ in $$[c,d]$$:

First, note that $$a+c\le r\le a+d$$ or $$a+d\le r\le b+d$$.

In the former case, there must be $$y$$ in $$[c,d]$$ such that $$r=a+y$$. Let $$x=a$$.

In the latter case, there must be $$x$$ in $$[a,b]$$ such that $$r=x+d$$. Let $$y=d$$.

Note: I'm not sure that this engages with OP's concerns. And I suspect that what I've written here has been written before on this site, probably multiple times. But it does seem to speak to some of the comments on the question.

• Sure you don't have some typos here? For example $a+c \le r \le a + d$? How can this be in all cases? Mar 22, 2021 at 16:01
• I didn't say it was true in all cases. I wrote $a+c\le r\le a+d$ OR $a+d\le r\le b+d$. Mar 23, 2021 at 9:01

Since none of the comments (nor the single answer) provide a "real world example" as the question wanted, I wanted to provide a simple one.

Let's say you are going to buy some apples from two stores. Store X will give you a minimum of $$a$$ apples, and a maximum of $$b$$ apples, thus its range is $$[a, b]$$. Meanwhile, store Y will give you a minimum of $$c$$ apples and a maximum of $$d$$ apples, thus its range is $$[c, d]$$.

In the worst case scenario, both stores will give you their minimum amount of apples. That would be $$a$$ apples from store X and $$c$$ apples from store Y. And in the best case scenario, they give you $$b$$ and $$d$$ apples, respectively. Thus, the new range that represents the sum of the number of apples that you can get from the two stores would be $$[a+c, b+d]$$.