4.3 Arbitrary Unions and Intersections

Objectives:

• Generalize the definition of unions and intersections to apply to infinite collections of sets

• Apply arbitrary unions and intersections to statements involving infinitely many sets

4.3.1 Reaching Into the Infinite

Example 4.3.1.

Consider the list of sets below. Let’s say we took all the elements of each of these sets $A_1,A_2,A_3,\mathrm{\dots }$ and threw them together into one big set - something like an infinite union of sets. What would this set look like? What numbers would it contain?

$$A_1=\{0\},A_2=[0,\frac{1}{2}],A_3=[0,\frac{2}{3}],\mathrm{\dots },A_n=[0,1-\frac{1}{n}],\mathrm{\dots }$$

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Now consider the following list of sets. What elements of $ℝ$ are in all of these sets? What would this “infinite intersection” look like?

$$B_1=[0,\frac{1}{2}),B_2=[0,\frac{1}{4}),B_3=[0,\frac{1}{8}),\mathrm{\dots },B_n=[0,\frac{1}{2^n}),\mathrm{\dots }$$

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Mathematics begins to defy our expectations when we start dealing with infinite numbers of sets. Truthfully, we do not have a good grasp of what an infinite number of objects looks like. There are not an infinite number of atoms in our universe, not an infinite number of seconds in our universe’s history, and not an infinite number of possibilities that can take place in any billions of years beyond this point in time.

But mathematics works with the infinite almost exclusively now - limits, sequences, and Taylor series are just a few examples of tools that need reaching into the infinite to work. Why do we not like these topics very much? Perhaps it is exactly because of this notion of infinity that we don’t understand very well. But these very topics are why mathematics is incredibly useful in computing huge amounts of data quickly and accurately. It is why we can be sure our actuarial methods are solid no matter how big the accounts, no matter how long we use them. We can do this all day… for an infinite amount of time… and mathematics will still hold firm.

Definition 4.3.2.

An index set $\mathrm{\Delta }$ is an arbitrary collection of elements where each element $i\in \mathrm{\Delta }$ has a set $A_i$ defined for that element.

Let ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Delta }}$ be a collection of sets for some index set $\mathrm{\Delta }$. Then the union of the entire collection of sets ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Delta }}$ is defined to be

$$\bigcup _{\alpha \in \mathrm{\Delta }}{A}_{\alpha }:=\{x\mid x\in A_{\alpha }\text{ for }\text{at least one }\alpha \in \mathrm{\Delta }\}.$$

We define the intersection of the entire collection of sets ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Delta }}$ to be

$$\bigcap _{\alpha \in \mathrm{\Delta }}{A}_{\alpha }:=\{x\mid x\in A_{\alpha }\text{ for }\text{all}\alpha \in \mathrm{\Delta }\}.$$

In the special case that $\mathrm{\Delta }=\mathbb{N}$, we write

$$\bigcup _{n=1}^{\mathrm{\infty }}{A}_n=\{x\mid x\in A_n\text{ for }\text{at least one}n\in \mathbb{N}\}$$

and

$$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n:=\{x\mid x\in A_n\text{ for }\text{all}n\in \mathbb{N}\}.$$

The LaTeX for these symbols: $\bigcup$ is $\backslash$bigcup, and $\bigcap$ is $\backslash$bigcap.

Example 4.3.3.

Let $A_n$ be the set containing the first $n$ letters of the modern English alphabet, for $n\in \{1,2,3,\mathrm{\dots },26\}$. Describe the following sets using roster notation:

1. ${\bigcup }_{n=1}^{13}{A}_n$

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2. ${\bigcap }_{n=1}^{26}{A}_n$

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Example 4.3.4.

Let for $n\in \mathbb{N}$. Describe the following using set-builder notation:

1. ${\bigcup }_{n=1}^{\mathrm{\infty }}{S}_n$

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2. ${\bigcap }_{n=1}^{\mathrm{\infty }}{S}_n$

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4.3.2 Proving with Arbitrary Collections of Sets

Example 4.3.5.

Let $A_1=\{1,2\}$, $A_2=\{2,3\}$, $A_3=\{3,4\}$, $\mathrm{\dots }$, $A_n=\{n,n+1\}$, $\mathrm{\dots }$.

Prove that ${\bigcup }_{n=1}^{\mathrm{\infty }}{A}_n=\mathbb{N}$.

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Prove that ${\bigcap }_{n=1}^{\mathrm{\infty }}{A}_n=\mathrm{\varnothing }$.

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Example 4.3.6.

Consider the following sets:

$$Q_1:=\{0,1\},Q_2:=\{0,\frac{1}{2},1\},Q_3:=\{0,\frac{1}{3},\frac{2}{3},1\},\mathrm{\dots }$$

$$Q_n=\{\frac{m}{n}:m\in \{0,1,2,\mathrm{\dots },n\}\},\mathrm{\dots }$$

Show that ${\bigcup }_{n=1}^{\mathrm{\infty }}{Q}_n=\mathbb{Q}$.

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What is the set ${\bigcap }_{n=1}^{\mathrm{\infty }}{Q}_n$ in roster or set-builder notation? Prove your answer.

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