# An elementary doubt

This doubt has been bothering me for ages. I had learnt that if we have 2 intervals $(a,b)\cup(c,d)$, we can merge these together to form $(a,d)$ as long as $a and $b>c$. However, I saw one of the teachers at my coaching institute do this: $(a,b)\ \cup(b,c)=(a,b)$ How can we merge these two intervals together? Was what I had learnt incorrect? And in general, when can we merge $2$ or $more$ intervals (of all 4 types: $(a,b],(a,b),[a,b),[a,b]$) together to form a single interval? Any help would be really, really appreciated! Thanks very, very much in advance!

Note by Ishan Dasgupta Samarendra
5 years, 4 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

@Brian Charlesworth Sir, @Michael Mendrin Sir, @Pranjal Jain  Sirs, please could you help me? Many thanks!  PS. I'm really sorry for bothering you.

- 5 years, 4 months ago

Yeah, that seems funny to me. If your teacher wrote

$[a,b] \cup [b,c] = [a,c]$

then that would have been fine, since $[a,b]$ is a closed interval, i.e., includes the end points. However, $(a,b)$ is an open interval, so you'd think that the point $b$ shouldn't be in the union.

I've always hated real analysis when I was in school.

- 5 years, 4 months ago

Yes, Sir, even I was surprised when I saw it was an open interval - I think this would be possible only for a closed interval.  Wow Sir, you were taught Real Analysis in your school! My Uncle did it first when he went to Swarthmore!

- 5 years, 4 months ago

Actually, I didn't have real analysis homework until I was in college, but I had read up on it long before then.

- 5 years, 4 months ago

Wish I had a millionth of your brain, Sir... Would never have to worry about anything then:)

- 5 years, 4 months ago

I'm assuming that what your teacher wrote was $(a,b) \cup (b,c) = (a,c),$ which would only be correct if $b = c,$ (which wouldn't make much sense anyway). Otherwise, as Michael has said, if $b \lt c$ then $b$ would have to be excluded from the interval "sum".

As for the other types of intervals, we would have both $(a,b] \cup (b,c) = (a,c)$ and $(a,b) \cup [b,c) = (a,c).$

- 5 years, 4 months ago

Yes, Sir, your assumption was right. Thanks so much for your help, Sir. I'm really, really grateful to you!

- 5 years, 4 months ago