## Definitions

A set is a collection of things. In mathematics, we deal primarily with sets of numbers; in Qabalah with letters or words. The most common sets in mathematics are the sets natural numbers and the integers.

In this case, we will be looking at sets of letters (the English Alphabet) and sets of Natural Numbers (1,2, 3,… etc). We define them as follows:

*Let Q be the set of English Letters, that is Q = {A, B, C,…, Z}.**Let N be the set of Natural Numbers. N = {1, 2, 3, 4, ... } = the natural numbers*^{3}.

These are the only two sets we will be interested in for this discussion.

Some definitions follow.

## Elemental Sets and Operations on Elemental Sets

A set is a collection of elements. Sets can be combined with other sets or elements. One can add to a set, and subtract from it. One can multiply two sets or multiply a set by a single element. One can in fact, define any sort of function or mapping one likes between two sets: that is, we can associate any two pairs (one from each) according to any rule or whim we choose.

*If A is a set, and x is an element in A, then we write: x ϵ A.*

For example, the English Alphabet is a set of letters. ‘b’ is a letter in the English Alphabet, therefore, b ∈ {English Alphabet} or b ∈ Q where Q is the set of English Letters.

*If A is a set, and x is not an element in A, then we write: x ∉ A.*

For example, the number 13 is an element of N, the natural numbers but not of Q the set of English Letters. Therefore, 13 ∉ Q, but 13 ∈ N.

## The Algebra of Sets

We introduce some of the basic notation and operations on sets.

Supposing A and B are sets, then

*If every element in A is also contained in B then we say that A is a subset of B (A ⊂ B**).*

Note : A and B are equal if and only if A ⊂ B and B ⊂ A.

*A ∪ B : The union of A and B, is the set of all elements that are either in A or in B or in both. *

*A ∩ B : The intersection of A and B is the set of all elements that are in both sets A and B. *

*A \ B: is the set of all elements from A that are not in B. *

*Finally, two sets are said to be disjoint if A ∩ B ** = 0 (the empty set).*

Note that Q and N are disjoint, the former being a set of letters and the latter one of numbers so that they share no common term between them.

Generally, the associative, commutative, and distributive laws of arithmetic hold true for operations upon Sets as well as operations upon their Elements. The foundation of mathematical science was laid in the simplicity of these three laws. Their importance should not be overlooked. We state them here for the Student’s consideration. We illustrate with the case of natural numbers (though these laws apply equally to sets) before proceeding with examples in English Gematria.

1. **The ****Associative Law** states that it matters not the order in which we sum things (if we are looking at the parts of a whole):

Associative law : (a + b) + c = a + (b + c) = a + b + c , ∀ a, b, c ∈ N.

e.g. a =4, b =8, c = 3

(a + b) + c = (4 + 8) + 3 = 12 + 3 = 15

a + (b + c) = 4 + (8 + 3) =4 + 11 = 15

a + b + c = 4 + 8 + 3 = 15

2. **The Commutative Law** states that in matters not what is first or last, right or left, when a union of the two (or more) is the aim and result:

Commutative law : a + b = b + a , ∀ a, b ∈ N

e.g.: 4 + 8 = 8 + 4

3.** The Distributive Law** states a law of immense practical importance : namely that the effect of one element’s combined operation upon a composite is equal to the sum of the separate effects of that element’s operation into the composite’s constituent parts:

Distributive law : a × (b + c) = (a × b) + (a × c) , ∀ a, b, c ∈ N.

e.g. a =4, b =8, c = 3

a × (b + c) = 4 × (8 + 3) = 4 × 11 = 44

(a × b) + (a × c) = (4 × 8) + (4 × 3) = 32 + 12 = 44

We find that though these laws hold in neater worlds of Arithmetic, Algebra and Calculus they do not always hold for physics and certainly not for Gematria.

In our case, if we take Q to be the set of English Glyphs, then the associative, and distributive laws hold for Q as well. The commutative law will not hold in the case of final values. For instance,

**Associative law** : (a + b) + c = a + (b + c) = a + b + c , ∀ a, b, c ∈ Q.

e.g.: a = “The” , b = “Law”, c = “Thelema”

(a + b) + c = (The + Law) + Thelema = 220 + 221 = 441

a + (b + c) = The + (Law + Thelema) = 106 + (114 + 221) = 441

a + b + c = The + Law + Thelema = 106 + 114 + 221 = 441

**Commutative law** : a + b = b + a ∀ a, b, c ∈ Q

e.g.: a = “The Name” , b = “BABALON”

a + b = (The Name) + Babalon = 221 + 156 (n final) = 377

b + a = Babalon + The Name = 220 + 221 = 441

So that, Babalon, the Name ≠ The Name Babalon

## Relations and Functions on Sets

The English key is a function (or a mapping) from the set of English Alphabet Glyphs to a subset of the Natural Numbers. This means that for the sets Q and N, there exists a collection of ordered pairs (q, n) such that q ∈ Q and n ∈ N. This is called a relation, and this relation and its ordered pairs is what is illustrated in table I – The English Key at the beginning of this book.

We denote this relation by the following: f(Q) ↦ N.

Mathematically, if Q and N are two sets, then a function f from Q to N is a relation between Q and N such that for each q ∈ Q there is one and only one associated n ∈ N. The set Q is called the domain of the function, N is called its range.

Using the English Key mapping function as an example, we have : f(e)=1, f(a)=4, f(me)=41, etc. and see that Q here is the set of Roman Script letter while N is its range; that is f maps Q into N.

We now introduce some theorems. First we show that for every number in existence, there exists at least one letter combination that will sum to that number, so that there is no number without expression.

#### Theorem I

*Let f be the mapping of the English Key denoted by f(Q)↦R where Q is the set of English Alphabet letters and R ⊂ N.*

*Then f is additive, that is *

*f(a + b)= f(a) + f(b), ∀ a, b ∈ Q*

** Proof**: by Induction (omitted here).

#### Theorem II

L*et N be the set of Natural Numbers and let f(Q)↦ R be the English Key mapping where Q is the set of English Alphabet letters and R ⊂ N, then the relation f(Q)↦R is a function.*

**Proof**

A function f from Q to N is a relation between its domain Q and its range N such that for each q ∈ Q there is one and only one associated n ∈ N.

If we take final and initial, soft and hard values for separate instances of the domain variable, then from table I, we can see that each glyph corresponds to one and only one number. The proof follows.

#### Theorem III

*Let N be the set of Natural Numbers and let f(Q)↦R be the English Key mapping where Q is the set of English Alphabet letters and R ⊂ N. Then for all n ∈ N, there exists at least one finite combination of q1, q2,…,qr ∈ Q, such that*

*f(q1,q2,…,qr) = f(q1)+ f(q2)+…+ f(qr) = n.*

**Proof.**

0 is an element of Q. E=1 so that Q contains the unit. The rest follows from induction.

f(0) = 0, f(e) = 1, f(qk+1) = f(qk) + f(e) = f(qk + e)

The principle of Mathematical Induction states that if S is a subset of N such that, (1) 1 ∈ S and, (2) if k ∈ S then k + 1 ∈ S is True, then S = N.

If we substitute S for R, where R ={ f(q1,q2,…,qr) }, the set of values attained from applying the transformation f(Q)↦R upon all possible letter combinations in Q; and let N be the Natural numbers, then we see that:

- R ⊂ N
- (e)=1 so that 1 ∈ R and the first condition is verified.
- If k ∈ R, then by definition, there exists some combination of q1,q2,…,qr ∈ Q such that f(q1,q2,…,qr) = k. We can always add the unit element f(e) = 1. By theorem I, we have k+1 = f(q1,q2,…,qr) + f(e) = f(q1,q2,…,qr, e) so that k + 1 ∈ R.

Therefore, R = N. Q.E.D.

#### Corollary I

*The elemental set Q of English Key glyphs spans the space of Natural Numbers, so that Q is a basis for N.*

This is essentially a restatement of Theorem II in algebraic terminology. Namely that the set of Q is sufficient to span the Natural Numbers: for any n ∈ N, there exists a set of coefficients a1, a2, a3…, al ∈ N such that a1f(q1) + a2f(q2) + a3f(q3) + … + ajf(qj) = n ∈ N, where aif(qi) is the value of the letter qi ∈ Q in the set of English glyphs.

In laymen’s terms, any positive integer can be expressed by a combination of letters and their corresponding numbers according to the English Key mapping.

1 It is important to note, that any composite number is in fact a finite product of primes, and can be expressed as such. These considerations are valuable when investigating the nature of a particular number.

2 Proof should be obvious to the student. Argue by contrapostive: Axiomatically, it is impossible to multiply the same list of numbers and arrive at different results.

3 “0” is sometimes included in the set of natural numbers.