How to read "$\bigcup_{n = 1}^\infty$", as in $\bigcup_{n = 1}^\infty[2n, 2n+1]$

Question

So I am given an equation such as this: $$ T = \bigcup_{n = 1}^\infty[2n, 2n+1] $$

And I am meant to find the infimum and supremum of the set however I was never taught how to read what this mathematical notation is saying? What will the set look like?

Answer 1

It indicates the union of all the closed intervals of the kind $[2n,2n+1]$ with $n\in \mathbb{N}$, that is

$$[2,3]\cup[4,5]\cup...\cup[2n,2n+1]\cup...$$

from here you can easily deduce infimum and supremum.

Answer 2

when $n=1, [2n, 2n+1]=[2,3]$

when $n=2, [2n, 2n+1]=[4,5]$

when $n=3,[2n, 2n+1]=[6,7]$

and et cetera.

$T$ is the union of closed intervals of length $1$, where the left boundary is an even number that is at least $2$.

In general $$\bigcup_i A_i = \{ x: \exists i, x \in A_i\}$$

Answer 3

The small numbers above and below the $\bigcup$ tells you what different values of $n$ we consider, and for each $n$ whatever is on the right of the symbol becomes a different set. So you have the collection $$ \underbrace{[2, 3]}_{n = 1}, \underbrace{[4, 5]}_{n = 2}, \underbrace{[6, 7]}_{n = 3},\ldots $$ And then the symbol $\bigcup$ itself tells you that you combine them using the union operation ($\bigcup$ is a large $\cup$, after all). There is a corresponding notation $\bigcap$ for intersections.

Compare this to something like $$ \sum_{n = 1}^\infty \frac1{n^2} $$ where the small numbers above and below $\sum$, as well as the expression to the right of $\sum$, together tells you that you have the set $$ \underbrace{\frac11}_{n = 1}, \underbrace{\frac14}_{n = 2}, \underbrace{\frac19}_{n = 3}, \ldots $$ of numbers, and then the symbol $\sum$ itself tells you what you should do with them (namely add them).