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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