# No, You First

### Commutativity

The point of commutativity is to swap the positions of two variables in a function. Normally, in algebra, this cue barging is only allowed in special functions that can be preserved through the process. In generalized truth function algebra commutativity is guaranteed, you just need to change the truth function.

Let's say you have some truth function, $(\textbf{A}^n)_\alpha$, and you want to swap $A_i$ and $A_j$ variables (for simplicity I've made substitutions, $X=A_i$ and $Y=A_j$).

$(A_0,\ldots,A_{i-1},\underline{X},A_{i+1},\ldots,A_{j-1},\underline{Y},A_{j+1},\ldots,A_{n-1})_\alpha=$

$=(A_0,\ldots,A_{i-1},\underline{Y},A_{i+1},\ldots,A_{j-1},\underline{X},A_{j+1},\ldots,A_{n-1})_\beta$

From here we have only to calculate $\beta$ bitwise, with the standard operators for $A_i$ and $A_j$ commuted.

$$\beta=[\sigma^n_0,\ldots,\sigma^n_{i-1},\underline{\sigma^n_{j}},\sigma^n_{i+1},\ldots,\sigma^n_{j-1},\underline{\sigma^n_{i}},\sigma^n_{j+1},\ldots,\sigma^n_{n-1}]_\alpha$$