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Set Theory for Game Developers

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What is a Set?

Arrays and dictionaries both represent data sets. A set is a collection of objects or elements.

YouTube Video: Set Theory for Game Developers Video

In mathematics we say if x is an object and S is a set, then x can either belong to or not belong to S.

This means the statement “x is an element of S” is a proposition, so it can either be true or false. But it can't be both.

If x is an element of S:

x ∈ S

If x is NOT an element of S:

x ∉ S

Now let's say set S is equal to all positive even numbers:

S = {2,4,6,8...}

If the above statement is true, then the following statements are also true:

1) 3 ∉ S
2) 100 ∈ S
3) 97 ∉S
4) 8 ∈ S

How Do You Define a Set?

There are two ways you can define a set.

1) By Extension: Defining a set by its values

✶ S = {1,8,7}
✶ Y = {2,4,6...}
✶ hatclan = {“Hatnix”, “Sir Diealot”, "Daisy"}

2) By Intention: Defining by membership conditions

✶ {x | x is an even number}
✶ {x | x is divisible by 3}
✶ {x | x > 42}

Equal Sets

Sets are said to be equal if they both contain the exact same elements.

X = {6,66,666}
Y = {6,66,666}

Set X is equal to set Y:

X = Y

If sets X and Y are equal and 6 is an element of X, then 6 must also be an element of Y. Similarly, if 42 is not an element in X, then it is not an element in Y.


If all of A's elements are also contained in B, then A is a subset of B:

A ⊆ B

If A is a subset of B and B is a subset of A, then A and B are equal sets:

(A ⊆ B) (B ⊆ A) A = B

Null Sets

If a set holds no values it is said to be empty or null:


A union B means all of the elements that are contained in A or B. If an element exists in A, it is included in this union. If x exists in B, it is also included in this union. If x exists in both A and B, it exists in this union.

Picture a man and woman getting married. The man comes from family A and the woman comes from family B. The union of family A and B is all the members of both families.

A union B: A ∪ B

(See the blue portion below)
Union of A and B


The intersection between A and B is equal to all elements found in A that also found in B. If group A is all Americans who speak Japanese and group B is all Americans who speak Spanish, the intercetion of groups A and B is all Americans who speak Spanish and Japanese.

A intersection B: A ∩ B

(See the purple portion below)
Intersection of A and B


A minus B equals all the elements in A that are NOT contained in B. Let's say Group A is American voters and Group B is Americans who voted for Donald Trump. Group A minus Group B is all the American voters who did Not vote for Donald Trump.

A minus B: A - B

(See the blue portion below)
A minus B

If you want to learn more about logic, check out Logic for Programmers. This page includes aditional logic videos and a logic quiz game!

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Because it is crucial to achieving upward economic mobility, Coding Commanders believes everyone should have access to free quality STEM education. My tutorials explain complex concepts in plain English using relatable examples. No previous mathematics or technical knowledge required!

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