# Set Theory

## Set Theory

### Sets

A **set** is a collection of things, which could be numbers. **Set Theory** is the study of sets and the properties they have. The things that make up a set are called the **elements** of the set.

You may not realize it, but you interact with lots of sets everyday. Any elements with something in common are part of a set. For example, fruits are a set. Bananas, apples, and blueberries are all elements of this set. You can probably think of hundreds of other everyday examples of sets.

Here is an example of a set with numbers, called set **A**:

**A** = {3, 8, 20}

The number of elements of **A** is a property of the set **A**.

**A** has three elements: the numbers **3**, **8** and **20**. We can also show that something is an element of **A** using the symbol **∈**. The symbol **∈ **is like saying “belongs to”.

So, for set **A**:

**3**∈**A**, **8**∈**A** and **20**∈**A**

We would say it as “3 belong to set A, 8 belongs to set A and 20 belongs to set A.”

An important thing about sets is that the order of the elements does not matter. For example:

**A** = {3, 8, 20} is the same set as **A** = {8, 3, 20}

Here are examples of other sets, from Mrs. Brown’s class

**B** = {all of the people with brown hair} = {Roberta, Jessica, Abdul, Yuto}

**C** = {all of the people with blue eyes} = {Riley, Josiah, Kobe, Roberta, Jessica}

How many elements are there in these sets?

**Question 1:** How many elements are in set B?

B =

(The answers are at the bottom of the page)

**Question 2:** How many elements are in set C?

C =

### Subsets

A **subset** is a set that is ‘contained’ within another set.

For example:

**D** = {**3, 8**} is contained in the set **A** = {**3, 8**, 20}

There is also a symbol to show that a set is a subset of another set. It is **⊆**. The symbol **⊆ **is like saying “is a subset of". So, for our example above:

**D** ⊆ **A** and {3, 8} ⊆ {3, 8, 20}

**Question 3:** What are the possible subsets found in the set {**1, 2, 3**}?

### Intersection

As you might guess, sometimes sets have some elements in common. The **intersection** of two sets is a set of all the elements that the two sets have in common. The symbol for intersection is **∩** . The symbol **∩** is like saying “and”. Let’s look at an example.

Let’s take the sets of people from Mrs. Brown’s class (remember sets **B** and **C**). The intersection of sets **B** and **C** would be written as **B ∩** **C**. It has all of the people who are in set **B** (all of the people with brown hair) who are also in set **C** (they also have blue eyes).

So:

**B ∩** **C** = {All of the people with brown hair **AND** blue eyes}

**Question 4:** Which students are **B∩C** in Mrs. Brown’s class?

Here is another example that uses numbers:

X = {0, 8, 2, 100}

Y = {1, 12, 50, 8}

Z = {8, 12, 1, 42, -1}

**Question 5:** What is **X∩Y**?

**Question 6:** What is **Y∩Z**?

### Union

Another interesting concept in Set Theory is **union**. The union of two sets is what you get when you combine two sets together. It's like adding them. But, if the sets have an element in common, for example if they both contain a 4, then the union of those sets will only have ONE 4, not two. The symbol for union is **∪**. The symbol **∪ **is like saying “and but no duplicates”.Look at the following example.

### {4, 11} **∪** {2, 11} = {2, 4, 11}

Notice how the **11** is written only once.

Let’s look at the sets in Mrs. Brown’s class again (sets **B** and **C**).

**B** = {all of the people with brown hair} = {Roberta, Jessica, Abdul, Yuto}

**C** = {all of the people with blue eyes} = {Riley, Josiah, Kobe, Roberta, Jessica}

The union of sets **B** and **C**, written **B ∪ C**, is the set of all the people who are in at least one of these sets; therefore, it is the people with brown hair, blue eyes or both brown hair and blue eyes. For union, each element (in this case person) is counted only once.

**B ∪** **C** = {All of the people with brown hair and all of the people with blue eyes and all of the people with both brown hair and blue eyes}

**B** = {all of the people with brown hair} = {**Roberta**, **Jessica**, **Abdul**, **Yuto**}

**C** = {all of the people with blue eyes} = {**Riley**, **Josiah**, **Kobe**, Roberta, Jessica}

**B ∪** **C **= {**Roberta**, **Jessica**, **Abdul**, **Yuto**, **Riley**, **Josiah**, **Kobe**}

Try these with the sets **X**, **Y** and **Z**.

X = {0, 8, 2, 100}

Y = {1, 12, 50, 8}

Z = {8, 12, 1, 42, -1}

**Question 7:** What is **X ∪** **Y**?

**Question 8:** What is **Y ∪** **Z**?

### Why is set theory important?

Set theory is seen as the foundation of all mathematics. Sets are the building blocks of mathematics. The language of sets can be used to describe all mathematical concepts. Linear algebra, graph theory, analysis, and number theory are all areas of math that are built around sets.

Set theory also plays an important role in computer programming. Sets and Boolean logic are used in many programming languages. For example, there are different types of variables in many programming languages. These types might include text, numbers, or **booleans**.

We can think about each type of variable as a set. In both computer science and math, set theory is a way of clearly expressing an abstract concept.

## ANSWERS

**Question 1:**

How many elements are in set **B**?

**B** = 4

**Question 2:**

How many elements are in set **C**?

**C** = 5

**Question 3:**

What are the possible subsets found in the set {**1, 2, 3**}?

{1} ⊆ {1, 2, 3}, {2} ⊆ {1, 2, 3}, {3} ⊆ {1, 2, 3}, {1, 2} ⊆ {1, 2, 3}, {1, 3} ⊆ {1, 2, 3}, {2, 3} ⊆ {1, 2, 3}

**Question 4:**

Which students are **B ∩** **C** in Mrs. Brown’s class?

**B** = {**Roberta**, **Jessica**, Abdul, Yuto}

**C** = {Riley, Josiah, Kobe, **Roberta**, **Jessica**}

**B ∩** **C **= {**Roberta, Jessica**}

**Question 5:**

What is **X ∩** **Y? **

X = {0, **8**, 2, 100}

Y = {1, 12, 50, **8**}

**X ∩** **Y** = {**8**}

*Note:** {8} is a set, even though it has only one element!*

**Question 6:**

What is **Y ∩** **Z? **

Y = {**1**, **12**, 50, **8**}

Z = {**8**, **12**, **1**, 42, -1}

**Y ∩** **Z** = {**1,** **8, 12**}

**Question 7:**

What is **X ∪** **Y?**

X = {**0**, **8**, **2**, **100**}

Y = {**1**, **12**, **50**, 8}

**X ∪** **Y** = {**0, 1, 2, 8, 12, 50, 100**}

**Question 8:**

What is **Y ∪** **Z?**

Y = {**1**, **12**, **50**, **8**}

Z = {8, 12, 1, **42**, **-1**}

**Y ∪** **Z** = {**-1, 1, 8, 12, 42, 50**}

## Learn More

The mathisfun.com website has an excellent introduction to set theory including different types of sets and **set notation** (all of those funny symbols). At the bottom of the page are practice questions.

## References

Stanford Encyclopedia of Philosophy. (2014, October 8). Set Theory.