# 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 $$a<d$$ 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
3 years, 5 months ago

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Your initial statement is correct.

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 $$(a,b) \cup [b,c) = (a,c).$$

- 3 years, 4 months ago

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

- 3 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.

- 3 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!

- 3 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.

- 3 years, 4 months ago

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

- 3 years, 4 months ago

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

- 3 years, 5 months ago