## 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}

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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.

• = {..., -3, -2, -1, 0, 1, 2, 3, ...}, the set of integers.
• = {..., -1/2, 0, 1/2, 1/3, 2/3, 1/4, 3/4, ...}, the set of rational numbers.
• = {..., 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.

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### 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

### Try Some Practice!

Sets, Elements, and Cardinality.pdf
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