Set notation

I was taught set notation when I was in sixth grade. The teacher started by giving us the definition of a set.

Basically, he said that anything with a collection of objects is a set. Having said that, we felt afterward that we could easily make up our own sets.

You could be surprised what a set could be made of. Basically, any collections of things that make sense.

Examples of sets:


The set of all letters in the modern English alphabet

What is included in that set? a, b, c, d, e, f, etc...


The set of all great mathematicians in the past

What is included in that set? We could mention with Carl Gauss, Isaac Newton, Einstein, Blaise Pascal, Euclid, Pierre de Fermat, etc..Sorry, I am not silly to list them all here!


The set of all positive numbers less than 10

What is included in that set? 1, 2, 3, 4, 5, 6, 7, 8, 9


The set of all types of sausages

What is included in that set? I am not sure I am knowledgeable enough in this area. I know Italian sausage and ???. What else? Did I miss something?


The set of all states in the United States

Oh Boy.Geography! You know what.I think you got the point. Let us move on to something else

Finite and infinite set:

A set is finite if you can list all its elements and infinite otherwise.

All sets described above are finite because you can list or count all their elements

However, among the five sets, one set can be turned into an infinite set with one small change. That set is set number 3

If I get rid of the word positive and say instead "the set of all numbers less than 10" the set in now infinite because you cannot count all those numbers less than 10.

Ways to define a set:

a. With a verbal description: All sets above are described verbally when we say, " The set of all bla bla bla "

b. A listing of all members separated by commas with braces ({ and }):

A listing of set 1 is written as: {a, b,c,d,e,f,....,z}

c. Set-builder notation:

Set 1 and set 4 can be written as { x / x is a letter of the modern English alphabet} and { x / x is a type of sausage}

{ x / x is a letter of the modern English alphabet} is read, " The set of all x such that x is a letter in the modern English alphabet

Set-builder is an important concept in set notation. You must understand it.

We use capital letters such as A, B, and so forth to denote sets

For example, you could let A be the set of all positive numbers less than 10.

We use the symbol Î to indicate that an object belongs to a set and the symbol Ï to indicate that an object does not belong to a set

For example, if A is the set of all positive numbers less than 10, then 2 ∈ A, but 12 ∉ A

A set that has no element is called empty set and is denoted by { } or ∅

For example, {x / x is a human being who have lived 10,000 years} is an empty set because it is impossible to find at least one human being who have lived so long

Two sets are equal if they have exactly the same element

For example, { x / x is a number between bigger than 1 and less than 5} and { 2, 3, 4} are equal sets

Subtleties with set notation:

Two sets are still equal even if the same element is listed twice

{ 2, 3, 4} and { 2, 3, 3, 4} are equal

The order of elements in sets does not matter

{ 2, 3, 4} = { 2, 3, 3, 4} = { 4, 3, 2}

This concludes the lesson about set notation

Introduction to Set Notation Quiz. Check how well you understand this lesson.

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