## Equivalence Classes and Equivalence Relations

10 is an arbitrary number that demonstrates this concept simply. For the function f(x)=(x)mod10, the output on the interval [0-20] is: **(0)mod10≅0; (1)mod10≅1; (2)mod10≅2; (3)mod10≅3; (4)mod10≅4; (5)mod10≅5; (6)mod10≅6; (7)mod10≅7; (8)mod10≅8; (9)mod10≅9; (10)mod10≅0; (11)mod10≅1; (12)mod10≅2; (13)mod10≅3; (14)mod10≅4; (15)mod10≅5; (16)mod10≅6; (17)mod10≅7; (18)mod10≅8; (19)mod10≅9; (20)mod10≅0.**

The ≅ symbol means congruent to instead of equals. After (10)mod10≅0, the output of the function repeats. A period is the length of an interval before the relation repeats. In this example, the period is of length ten. The number of times entities repeat in a period is the frequency. The frequencies of magical spells created from this list are the number of times magical symbols occur for a mod. The output of the function means every integer is or is not a multiple of 10. If it is a multiple of 10, the remainder will be 0. If it is not a multiple of 10, it can be off by 1, 2, 3, 4, 5, 6, 7, 8, or 9. Consequently, all integers can be grouped according to their relationship to ten. The pairs 1 and 11, 2 and 12, or 3 and 13 can be grouped in sets together by their remainders. Each of the ten sets has an infinite number of integers to be grouped. For example, (3915000000000001)mod10≅1, so 1, 11, and 3915000000000001 can be put in a set, together. Algebraically, rules come from axioms, relations, and rules form formulas, so we can construct formulas if we construct axioms, and we can construct axioms by creating relationships. The relation comes from property that the difference of any two integers within one of those sets will be congruent to 10. For example, (3761)mod 10 and (2411)mod10 are both 1, so we can group them in the same set. 3761-2411=1350. 1350/10=135, where 1350mod10≅0. That congruence means that 3761 relates to 2411. That relation is 3761~2411. Since there is a relationship, this means there is a rule. These are examples of rings. A ring is an abstract structure formed by binary operations operating on two sets, such as addition or subtraction.

The grouping of integers is based on how the integers relate to each other. Group 1 is a grouping of all integers that relate to 1. Group 2 is a group of all integers that relate 2. Group 3 is a group of all integers that relate to 3. Those grouping of integers by the integer they relate to have three properties; they are reflexive, symmetric, and transitive. If a relation has all those properties, it is an equivalence relation. The set of all integers that relate to an integer is called an equivalence class. People can do computations and Algebraic manipulations with equivalence classes.

## Building A List of Magical Classes

Equivalence relations, modulo operations, and equivalence classes are very abstract concepts; however, these abstract concepts can be applied to create magical ontologies. Because there is an infinite number of symbols and possible rules, and the purpose of this is to introduce the concept, examples in this post are generic. See what you can make and do with it and be creative!

For n integer, the highest value of the remainder is n-1. I will use the number 10 for this example. The output for f(x)=(x)mod10 will give you 0-9 before it repeats, so there are ten possible groups. Because the periods are ten elements long, the mod is 10. The association of elements to an ordered set of integers creates a finite enumeration of the symbols known as a tuple or an alphabet. To emphasize the arbitrariness of magical ontologies, I am going to build a list of magical symbols from spell classes featured in Dragon Age and Elder Scrolls **[Magic, Creation, Form, Alteration, Reality, Conjuration, Illusion, Normality, Entropy, Destruction]**. It would be best if you considered how element at position n1 relates conceptually to all the other elements since you can construct a list of magical elements iteratively. Because of iteration, all the other elements are an induction that adds 1 to the k number in the sequence. 0 would be present in all the magical elements since you can write any position as n+0=n, so you should also consider how the 0th element relates to all the other ones. You want to think about how the system flows. Iterating over the list of magical symbols starting from 0 creates the following table:

## Magical Classes

Integer | Class |
---|---|

0 | Magic |

1 | Creation |

2 | Form |

3 | Alteration |

4 | Reality |

5 | Conjuration |

6 | Illusion |

7 | Normality |

8 | Entropy |

9 | Destruction |