## Sets, Elements, and Cardinality

### Sets

A set is a collection of objects known as elements. Objects can be real, like people, places, or things, or they can be abstract, like numbers or variables. An example of a set would be all the numbers between 1 and 10.

- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

We enclose our set in { } curly braces. Each element is listed inside the set with a comma in-between. Normally, we name our sets. We can give them real names or arbitrary capital letters as names. For example, here's a set of math courses at my local university that cover calculus courses.

- Calc = {MATH151, MATH152, MATH251, MATH252}

I chose to name it Calc, but I could have named it anything, like P, Q, C, A, etc.

### Elements

If an object is in our set, we say that that object is an element of the set, and we use the ∈ symbol. For example, suppose we have the following set of primary colors.

- P = {red, blue, yellow}

Since *yellow *is in our set, we can write* yellow **∈ P*. If an element is not in a set, we use the ∉ symbol. Since *green *is not a member of the set *P, *we would write *green *∉ *P.***

Elements are particular. That is, the element *'2'* and the element *2* are different elements.

- red ∈ P
- "red" ∉ P

Sets with exactly one element are called singletons.

- {1}

There are two conventions for sets that are important to keep in mind.

- The order of the elements in the set do not matter.
- {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}

- Repeated elements are not counted twice.
- {a, a, a, a, a, a, b, b, b, b, c, c, c, c, c} = {a, b, c}

### Infinite and Finite Sets

Sets can have a finite amount of objects or an infinite amount of objects. We've seen a few examples of finite sets above. Now, we can look at a couple common infinite sets.

- ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}, the set of integers.
- ℚ = {..., -1/2, 0, 1/2, 1/3, 2/3, 1/4, 3/4, ...}, the set of rational numbers.
- ℝ = {..., 0, 1, 2, e, π, ...}, the set of real numbers.

### Cardinality

We say that the cardinality of a set is how many elements the set has. We use the | | around the set to denote cardinality. For example, consider the three sets below:

- A = {1, 2, 3}
- B = {a, b, c, d, ..., x, y, z}
- C = {2, 4, 6, 8, ...}

This means that:

- |A| = 3, because it has 3 elements.
- |B| = 26, because it contains all of the English alphabet
- |C| is infinite, because it has an infinite number of elements.

### The Empty Set

When a set has no elements it is called the empty set. It is written as such:

- ∅ = { }

The cardinality of the empty set, |∅|, is 0.

### Sets in Sets

Sets can have other sets as elements. We treat the embedded set as a single distinct element.

- K = {x, y, {z}}
- |K| = 3

- P = {{a, b, c, d, e, f, g}}
- |P| = 1

- L = {∅}
- This set is equivalent to {{ }}
- |L| = 1

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