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Produced by Charles Wells     Revised 2017-02-25
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# SOME SPECIFIC SETS

## Sets of numbers

The following notation for sets of numbers is fairly standard.

### Remarks

• Some authors use $\mathbb{I}$ for $\mathbb{Z}$, but $\mathbb{I}$ is also used for the unit interval.
• Many authors use $\mathbb{N}$ to denote the nonnegative integers instead of the positive ones.
• To remember $\mathbb{Q}$, think "quotient".
• $\mathbb{Z}$ is used because the German word for "integer" is "Zahl".

Until the 1930's, Germany was the world center for scientific and mathematical study, and at least until the 1960's, being able to read scientific German was required of anyone who wanted a degree in science. Now, some German universities teach some high-level courses in English.

## Intervals

### Definition: interval

Let $a$ and $b$ be real numbers.

• The open interval $(a,b)$ is the set $\left\{ x\in \mathrm{\mathbb{R}}\,|\,a\lt x\lt b \right\}$.
• The closed interval $\left[ a,b \right]$ is the set $\{x\in\mathrm{\mathbb{R}}\,|\,a\le x\le b\}$
• The unit interval is defined to be the closed interval $[0,1]$.

#### Warning

The notation $\left( a,b \right)$ is also used to mean the ordered pair with first coordinate $a$ and second coordinate $b$.

### Picturing intervals

You can picture the set of real numbers as a line infinitely long in both directions.  Numbers $a$ and $b$ may be pictured as points on the line, this way:

The open interval $\left( a,b \right)$ is the set of points between $a$ and $b$, not including $a$ and $b$. The closed interval $\left[ a,b \right]$ is the set of points between $a$ and $b$, including $a$ and $b$.

## The empty set

### Definition: empty set

The existence and uniqueness of the empty set follows directly from the specification for setsThe empty set is completely determined by the fact that it has no elements.

#### Examples

• $\left\{x\in \mathrm{\mathbb{R}}\,|\,{{x}^{2}}\lt0 \right\}=\varnothing$, because every square of a real number is nonnegative. If something about this example bothers you look at unnecessarily weak assertion.
• If $a\lt b$, the open interval "$(a, b)$" defines the empty set. For example, $\left( 3,2 \right)=\varnothing$. That is because by definition, $\left( 3,\,\,2 \right)=\left\{ x\in \mathrm{\mathbb{R}}\,|\,3\lt x\lt 2 \right\}$, but there are no real numbers that are bigger than $3$ and less than $2$.
• It is also true that the closed interval $\left[3,2\right]=\varnothing$.

### Empty set as element of a set

Since the empty set is a set, it can be an element of another set

#### Example

Consider this: although "$\varnothing$" and "$\{\, \}$" both denote the empty set, $\left\{ \varnothing \right\}$ is not the empty set; it is the set whose only element is the empty set. The set $\left\{ \varnothing \right\}$ is in fact a singleton set.

#### Example

The set of subsets of the two-element set $\{1,2\}$ is the set $\{\varnothing,\{1\},\{2\},\{1,2\}\}$

#### More about "only one empty set"

Because there is only one empty set, $\left\{ x\in \mathrm{\mathbb{R}}\,|\,{{x}^{2}}\lt 0 \right\}$, $[3, 2]$ and $(3, 2)$ are all exactly the same set. The two statements below say this in two different ways. I recommend that you think about them until you understand that they are really saying the same thing.

• $\left\{ x\in \mathrm{\mathbb{R}}\,|\,{{x}^{2}}\lt 0 \right\}=[3,2]=(3,2)$.
• "$\left\{ x\in \mathrm{\mathbb{R}}\,|\,{{x}^{2}}\lt 0 \right\}$", “$[3, 2]$” and “$(3, 2)$”  are three different representations of the same set, namely the empty set.

### Metaphors and images for the empty set

The empty set does not get along very well with the way you think about bunches of things. The following examples give useful illustrations of how a math idea (empty set in this case) may not fit your intuition.

#### Container

If you think of a set as a container, the empty set is simply an empty container.  The only problem with this is that there is only one empty set, whereas normally you expect that any number of different containers can be empty.

#### Collection

If you think of a set as a collection, there is a problem: If I didn’t own any chess pieces, I would not talk about “my collection of chess pieces”. If you don’t have something, you don’t have a collection of them!

The concept of collection also has the same problem that “container” has:  You would mean two different things by the statements

• I don’t have a collection of chess pieces.
• I don’t have a collection of Hummel figures.

But there is only one empty set. It does not come in types!

#### Pointer

If you think of a set as a pointer to its elements, then the empty set is the same as the null pointer.  Again there is a problem: A computer program can have many different pointers set to null at the same time.  But there is only one empty set.

As always, images and metaphors for the concept of set produce cognitive dissonance.  Remember:

The definition always overrides your intuition

It would be perfectly possible to use a set theory with more than one empty set; for example, to agree that the empty set of integers is different from the empty set of continuous functions. In effect, that is what happens in certain versions of type theory. But the only-one-empty-set point of view is essentially standard in most math texts.

## Singleton sets

### Definition:singleton

A set containing exactly one element is called a singleton set.

#### Examples

• The singleton set $\{3\}$ is the set whose only element is 3.
• The singleton set $\{\{3\}\}$ is the set whose only element is $\{3\}$.
• The number $3$ is not a singleton set. It is not a set at all, it is a number.
• $\left\{\varnothing\right\}$ is the singleton set whose only element is the empty set.
• The closed interval $[3, 3]$ is the singleton set $\{3\}$, but the open interval $(3, 3)$ is the empty set.

### Remark

Because a set is distinct from its elements, a set with exactly one element is not the same thing as the element. Thus $\{3\}$ is a set, not a number, whereas $3$ is a number, not a set.