## Jesse's Girl

#### Associativity

$$\big(\textbf{B}^m,(X,\textbf{A}^n)_\alpha,Y,\textbf{C}^p\big)_\beta=\big(\textbf{B}^m,X,(\textbf{A}^n,Y)_\gamma,\textbf{C}^p\big)_\delta$$With associativity we can move variables out of one function and into another. Unfortunately, not even allowing truth functions to change will guaranty that associativity will work for a pair of functions.

This section explores the situations where associativity is possible and what truth functions will result.

### A Story Begins

$$\big(\textbf{B}^m,(X,\textbf{A}^n)_\alpha,Y,\textbf{C}^p\big)_\beta=\big(\textbf{B}^m,X,(\textbf{A}^n,Y)_\gamma,\textbf{C}^p\big)_\delta\ \Leftrightarrow\ \big((X)_{\alpha_i},Y\big)_{\beta_j}=\big(X,(Y)_{\gamma_i}\big)_{\delta_j}\ \forall\ i,j$$First thing's first. This section breaks down the general form of associativity to one involving only two variables and semi-parses of the component and principal operators.

### Is She The One?

Unlike commutativity, associativity is not always possible. In many cases there are no values for $\gamma$ and $\delta$ that produce the same function as $\alpha$ and $\beta$. Here we'll laboriously show where $\delta$, and by extension $\gamma$, are not possible.