Indexed sets
Summation notation
“Recall” that we can write a long sum of a bunch of numbers using summation notation.
\[ a_1+a_2+\cdots+a_n = \sum_{i=1}^{n} a_{i} \]
We can even write infinite sums:
\[ 1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{i}}+\cdots = \sum_{i=1}^{\infty}\frac{1}{2^{i}} \]
Indexed sets
Suppose we have a bunch of sets \(A_1,A_2,\ldots, A_n\). Then we can write:
\[ A_1\cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^{n} A_{i} \]
and
\[ A_1\cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^{n} A_{i} \]
Indexed sets
If \(A_1,A_2,\ldots, A_n\) are all sets, then
\[ \bigcup_{i=1}^{n} A_{i} = \{x : x\mathrm{\ belongs\ to\ at\ least\ one\ set\ }A_{i}\} \]
- \(A_{1}=\{1,4,10,12\}\)
- \(A_{2}=\{5,12,15\}\)
- \(A_{3}=\{1,4,15,35\}\)
What is \(\bigcup_{i=1}^{3}A_{i}\)?
Indexed sets
\[ \bigcap_{i=1}^{n} A_{i} = \{x : x\mathrm{\ belongs\ to\ every\ set\ }A_{i}\} \]
- \(A_{1}=\{1,4,10,12\}\)
- \(A_{2}=\{5,12,15\}\)
- \(A_{3}=\{1,4,15,35\}\)
What is \(\bigcap_{i=1}^{3}A_{i}\)?
Indexed sets
One can also take the union and intersection of infinitely many sets.
\(\bigcup_{i=1}^{\infty} A_{i}\) and \(\bigcap_{i=1}^{\infty} A_{i}\).
The first means the elements in at least one of the sets; the second means the elements in every set.
Example. For each \(i\in\mathbb{N}\), let
\[A_{i}=\{-i,0,i\}\]
What is \(\bigcup_{i=1}^{\infty} A_{i}\) and \(\bigcap_{i=1}^{\infty} A_{i}.\)?
Index sets
Instead of numbering the sets, one can label them with elements of any set \(I\) called an index set.
\(\bigcup_{i\in I} A_{i}\) is the set of elements that belong to at least one of the sets \(A_{i}\).
\(\bigcap_{i\in I} A_{i}\) is the set of elements that belong to every one of the sets \(A_{i}\).
Index sets example
Let \(C\) be the set of Counties in the state of Connecticut (there are 8 of these). For each county \(c\in C\), let \(T(c)\) be the set of Towns in that County.
For example, if \(c\) is Tolland County, then the elements of \(T(c)\) are Andover, Bolton, Columbia, Coventry, Ellington, Hebron, Mansfield, Somers, Stafford, Tolland, Union, Vernon, and Willington.
What is \(\bigcup_{c\in C} T(c)\)?
Index sets
Let \[ \mathbb{R}_{+} = \{r: r\in \mathbb{R}, r>0\}. \]
For every real number \(r\in \mathbb{R}_{+}\), let \[ A_{r} = \{(x,y)\in \mathbb{R}^{2} : x^2+y^2<r^2\}. \]
Index sets
What is \(\bigcap_{r\in\mathbb{R}_{+}} A_{r}\)?
Index Sets
What is \(\bigcup_{r\in\mathbb{R}_{+}} A_{r}\)?
Example
What is \(\bigcap_{i\in\mathbb{N}} [0,i+1]\)?
Example
Suppose that \(I\) and \(J\) are sets, that \(J\not=\emptyset\), and that \(J\subseteq I\). Is \[ \bigcap_{a\in I}A_{a}\subseteq\bigcap_{a\in J}A_{a}? \] Explain.