# Set (Card Game) and its Algebraic Representation

*Brian Li*

*2020-06-16*

*MAT388 Card Game Set Mathematics Algebra*

### # Introducing a card game called "SET"

"Set" is a game that is quite popular within the mathematics community. The game is played with a deck of **shapes** (one, two, three), the **colors** itself (oval, squiggle, diamond), the **number of shapes** (solid, unfilled, striped), and the **shading** (red, green, purple). The cheat sheet below created by Quanta Magazine offers a detailed explanation on how to play the game. You can also play around with the game here. At the end of this post, we will count the total number of SETs."

### # Algebraic Representation of the SET Deck

We want to label each cards with an algebraic representation. There are various motivation behind doing so. One is that giving algebraic representation to the deck allows us to program the game easily. Another reason is that having an algebraic representation allows mathematicians to compute the deck's probabilistic and combinatorics properties. Since there are

**Definition 1.0:** For a card in the game of set, the card can be expressed as a

shapes | colors | number of shapes | shading |
---|---|---|---|

oval | red | one | solid |

squiggle | green | two | unfilled |

diamond | purple | three | striped |

**Example 1.1:** Express the following cards algebraically.

- : the card has oval shape (
), color green ( ), two shapes ( ), and unfilled shading ( ). Therefore this card correspond to the tuple . - : the card has squiggle shape (
), color red ( ), three shapes ( ), solid shading ( ). Therefore the card corresponds to the tuple . - : the card has diamond shape (
), color purple ( ), one shape ( ), striped shading ( ). Therefore the card corresponds to the tuple .

### # The Set Product of the SET Deck

Now we will define an operation on

**Definition 1.2:** Let

**Theorem 1.3:** Given

**Proof:** To show that

If

, then , therefore . So the th feature of all three cards are all the same. If

, let us first assume that , then , then . This implies that , which contradicts the assumption. This shows that , thus the th feature of all three cards are all different. Similar results can be obtained when assuming .

These two cases above shows that the

Now we want to explore some algebraic properties of the set

**Lemma 1.4:**

**Proof:** Given two cards

**Lemma 1.5:**

**Proof:** We will prove this by providing a counterexample. Let

, so , so

We can see that

Now we are going to discuss some special properties of the Set Product.

**Lemma 1.6:** If

**Proof:** Let

**Theorem 1.8:** If

**Proof:** By Lemma 1.6,

Now it is clear that the set

**Definition 1.9:** We define **D-set** to be a subset

**Theorem 2.0:** There are in total

**Proof:** There are

**Theorem 2.1:** Given card

**Proof:** Once the card

### # References

[1] Paola Y. Reyes, The Mathematics of the Card Game Set, Digital Commons @ Rhode Island College

[2] Judy A. Holdener BS and MS and PhD (2005) PRODUCT-FREE SETS IN THE CARD GAME SET, Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15:4, 289-297, DOI: 10.1080/10511970508984123