Studies of Euler diagrams/transformations

These are pairs of functions in the same clan (NP equivalence class), so they can be expressed in terms of each other.
The clan numbers refer to the rational ordering. (Which will at some point be replaced by a better one.)

The transformation from one to the other is a Template:W, which means that arguments are negated and permuted.
It can be just a set of negated places or just a permutation. (These cases are marked with N, P or NP respecitively.)

Template:Anchorclan 84: Template:Boolfname and Template:Boolflink   (NP)

Template:Collapsible START This Euler diagram has no symmetry. Therefore the transformation of one into the other is unique.
The one from left to right is $\left(\begin{array}{cccc}0& 1& 2& 3\\ \mathrm{\neg }2& 0& \mathrm{\neg }1& 3\end{array}\right)$.   It means the following:

• The old red border becomes the new blue border, and the orientation changes. (The spikes pointed upward, now they point downward.)
• The old green border becomes the new red border. (The orientation stays the same.)
• The old blue border becomes the new green border, and the orientation changes. (The spikes pointed left, now they point right.)
• The yellow border remains unchanged.

Template:Boolfname

Template:Boolfname

The transformation from right to left is $\left(\begin{array}{cccc}0& 1& 2& 3\\ 1& \mathrm{\neg }2& \mathrm{\neg }0& 3\end{array}\right)$, the inverse of the one shown above. Template:Studies of Euler diagrams/transformations/dakota tinora/dh Template:Collapsible END

Template:Anchorclan 157: Template:Boolflink and Template:Boolflink   (NP)

Template:Collapsible START Template:Boolfname is a gap variant of Template:Boolfname (shown above on the right).

These two functions do not have the same set of (relevant) arguments. But that is not a problem.
One can see Template:Boolfname as a 5-ary function with an irrelevant input E, and Template:Boolfname has an irrelevant input A.

The transformation from left to right is $\left(\begin{array}{ccccc}0& 1& 2& 3& 4\\ \mathrm{\neg }3& 2& 1& 4& 0\end{array}\right)$. It means the following:

• The old red border becomes the new yellow border, and the orientation changes. (The spikes pointed left, now they point right.)
• The old green border becomes the new blue border.
• The old blue border becomes the new green border.
• The old yellow border becomes the new brown border.
• The inexistent old brown border "becomes" the inexistent new red border. (There could also be a negator in this place.)

Template:Boolfname

Template:Boolfname

The transformation from right to left is $\left(\begin{array}{ccccc}0& 1& 2& 3& 4\\ 4& 2& 1& \mathrm{\neg }0& 3\end{array}\right)$, the inverse of the one shown above. Template:Studies of Euler diagrams/transformations/dagoro darimi/dh Template:Collapsible END

Template:Anchorclan 109: Template:Boolfname and Template:Boolflink   (NP)

Template:Collapsible START The diagrams in this EC are mirror symmetric. That means that there are two transformations between each pair of functions.
The one used here to get from left to right is $\left(\begin{array}{cccc}0& 1& 2& 3\\ \mathrm{\neg }2& 3& 1& 0\end{array}\right)$. The one from right to left is the inverse $\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& 2& \mathrm{\neg }0& 1\end{array}\right)$.

Template:Boolfname

Template:Boolfname

Template:Anchorclan 203: Template:Boolflink and Template:Boolfname   (NP)

Template:Collapsible START Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for Template:Boolfname.
The transformation from the chosen diagram of Template:Boolfname to the one of Template:Boolfname above is $\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& 3& \mathrm{\neg }2& \mathrm{\neg }1\end{array}\right)$. Its inverse is $\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& \mathrm{\neg }3& \mathrm{\neg }2& 1\end{array}\right)$.

Template:Boolfname	Template:Boolfname
	Template:Boolfname

The transformation from the chosen diagram of Template:Boolfname to the one of Template:Boolfname below is $\left(\begin{array}{cccc}0& 1& 2& 3\\ \mathrm{\neg }1& \mathrm{\neg }2& 3& 0\end{array}\right)$. Its inverse is $\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& \mathrm{\neg }0& \mathrm{\neg }1& 2\end{array}\right)$.

This whole NP equivalence class with 192 functions (represented by 384 diagrams) can be found in [[Studies of Euler diagrams/dukeli NP|Template:Boolfname NP]]. Template:Studies of Euler diagrams/transformations/dukeli netuno/dh Template:Collapsible END

Template:Anchorclan 349: Template:Boolfname and Template:Boolflink   (P)

Template:Collapsible START These diagrams have the symmetry of a rectangle, which can be flipped in four ways.
This means that there are four transformations between each pair of functions in it.
The one used here to get from left to right is $\left(\begin{array}{cccc}0& 1& 2& 3\\ 1& 2& 3& 0\end{array}\right)$. The one from right to left is the inverse $\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& 0& 1& 2\end{array}\right)$.

The arguments are only permuted, but not negated. So only the colors change, but not the direction of the spikes.

Template:Boolfname

Template:Boolfname

Template:Anchorclan 15: Template:Boolflink and Template:Boolfname   (NP)

Template:Collapsible START Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for Template:Boolfname.
The transformation between the chosen diagram of Template:Boolflink and the one of Template:Boolfname above is $\left(\begin{array}{ccc}0& 1& 2\\ \mathrm{\neg }1& \mathrm{\neg }0& 2\end{array}\right)$. (It is self-inverse, so it works left to right, and back.)

Template:Boolfname	Template:Boolfname
	Template:Boolfname

The transformation from the chosen diagram of Template:Boolflink to the one of Template:Boolfname below is $\left(\begin{array}{ccc}0& 1& 2\\ \mathrm{\neg }2& 0& 1\end{array}\right)$. Its inverse is $\left(\begin{array}{ccc}0& 1& 2\\ 1& 2& \mathrm{\neg }0\end{array}\right)$. Template:Collapsible END

Template:Anchor Template:Boolfname and Template:Boolflink   (N)

Template:Collapsible START These two diagrams differ only in the orientation of border A. The self-inverse transformation between them is $\left(\begin{array}{ccc}0& 1& 2\\ \mathrm{\neg }0& 1& 2\end{array}\right)$.

Template:Boolfname

Template:Boolfname

Template:Anchor Template:Boolfname and Template:Boolflink   (P)

Template:Collapsible START The transformation between the two diagrams is $\left(\begin{array}{ccccc}0& 1& 2& 3& 4\\ 0& 3& 4& 1& 2\end{array}\right)$, which is self-inverse.
From left to right the columns 3 and 4 are irrelevant, and from right to left the columns 1 and 2.
The entries in these columns could be negated and permuted. Counting these leads to 8 possible transformations.
(That is between the two diagrams. Between the two functions it would be 16.)

Template:Boolfname

Template:Boolfname

Template:Anchorclan 19: Template:Boolflink and Template:Boolfname   (N)

Template:Collapsible START Euler diagrams in this EC have 3-fold Template:W. (This is obfuscated by the conventional representation on the right.)
Thus there are six transformations from one function to another.
The simplest one between these two is to change the orientation of border A, i.e. $\left(\begin{array}{ccc}0& 1& 2\\ \mathrm{\neg }0& 1& 2\end{array}\right)$.   (Compare potero and potula.)

Template:Boolfname

Template:Boolfname

Template:AnchorTemplate:Boolflink filtrates   (P)

Template:Boolflink and Template:Boolflink

Template:Boolflink and Template:Boolflink

Template:Boolflink and Template:Boolflink

Template:Anchorbloatless alternatives   (N)

Template:Boolflink

Template:Boolflink