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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
2 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 \((a,b) \cup [b,c) = (a,c).\)

Brian Charlesworth - 2 years, 5 months ago

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

Ishan Dasgupta Samarendra - 2 years, 5 months ago

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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.

Michael Mendrin - 2 years, 5 months ago

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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!

Ishan Dasgupta Samarendra - 2 years, 5 months ago

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

Michael Mendrin - 2 years, 5 months ago

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

Ishan Dasgupta Samarendra - 2 years, 5 months ago

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

Ishan Dasgupta Samarendra - 2 years, 5 months ago

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