👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

XNOR and XOR
👁 Image 👁 Image

👁 Image

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XNOR and XOR
👁 Image 👁 Image

Are operations relations in a 1-element universe?
👁 Image 👁 Image

1-element universe:

👁 Negative statements combined by AND

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

2-element universe:

👁 Negative statements combined by AND

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

3-element universe:

👁 Negative statements combined by AND

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

4-element universe - first example with 15 minterm relations:

👁 Negative statements combined by AND

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

5-element universe:

👁 Negative statements combined by AND (zoom in)

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND (zoom in)

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations (zoom in)

Usually the question is, if somewhere are no or some elements.
But one may also ask, if somewhere is an even or an odd number of elements.

Parity relations have a Hadamard pattern 👁 Image
where the others have a Sierpinski triangle 👁 Image
.

In a 1-element universe even means 0, and odd means 1:

👁 Negative statements combined by AND

👁 {\displaystyle ~\land }

👁 Affirmative statements combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

2-element universe:

👁 Even combined by AND

👁 {\displaystyle ~\land }

👁 Odd combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

3-element universe:

👁 Even combined by AND

👁 {\displaystyle ~\land }

👁 Odd combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

4-element universe:

👁 Even combined by AND

👁 {\displaystyle ~\land }

👁 Odd combined by AND

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations

5-element universe:

👁 Even combined by AND (zoom in)

👁 {\displaystyle ~\land }

👁 Odd combined by AND (zoom in)

👁 {\displaystyle ~\Leftrightarrow }

👁 Minterm relations (zoom in)

In 👁 A is an even number of elements.
👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 A is an even number of elements. 👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 A is an even number of elements. 👁 Image 👁 {\displaystyle \lor ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image

In 👁 A is an odd number of elements.
👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 A is an odd number of elements. 👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 A is an odd number of elements. 👁 Image 👁 {\displaystyle \lor ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image

👁 Image

👁 {\displaystyle \oplus ~}

👁 Image

👁 {\displaystyle \Leftrightarrow ~}

👁 Image

This is a different way to write the same:

In 👁 symmetric difference of A and B is an odd number of elements.

👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 symmetric difference of A and B is an odd number of elements. 👁 Image 👁 {\displaystyle \land ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image 👁 {\displaystyle \Rightarrow } In 👁 symmetric difference of A and B is an odd number of elements. 👁 Image 👁 {\displaystyle \lor ~} 👁 Image 👁 {\displaystyle \Leftrightarrow ~} 👁 Image

This is what the exclusive or excludes:

👁 Image

It's not to be confused with:

👁 Image

Just another example:

👁 Image

👁 Image

👁 {\displaystyle A\subseteq B}	👁 {\displaystyle \land ~}	👁 {\displaystyle B\subseteq C}	👁 {\displaystyle \Leftrightarrow }	👁 {\displaystyle A\subseteq B\subseteq C}
👁 Image 👁 Image 👁 Image 1011 1011	👁 {\displaystyle \land ~}	👁 Image 👁 Image 👁 Image 1100 1111	👁 {\displaystyle \Leftrightarrow ~}	👁 Image 👁 Image 👁 Image 1000 1011

There are 256 relations of this kind (corresponding to the 256 operations).
The 22 relations in the following table are shown in place of their mirrorings and rotations:

no and 1 one 👁 Image 👁 Image 👁 Image 0000 0000 👁 Image 👁 Image 👁 Image 👁 Image 0000 0001 👁 Image	7 and 8 ones 👁 Image 👁 Image 👁 Image 0111 1111 👁 Image 👁 Image 👁 Image 👁 Image 1111 1111 👁 Image
2 ones 👁 Image 👁 Image 👁 Image 1000 0001 👁 Image 👁 Image 👁 Image 👁 Image 0000 0101 👁 Image 👁 Image 👁 Image 👁 Image 0010 0001 👁 Image	6 ones 👁 Image 👁 Image 👁 Image 0111 1011 👁 Image 👁 Image 👁 Image 👁 Image 0101 1111 👁 Image 👁 Image 👁 Image 👁 Image 0111 1110 👁 Image
3 ones 👁 Image 👁 Image 👁 Image 0001 0110 👁 Image 👁 Image 👁 Image 👁 Image 1000 0101 👁 Image 👁 Image 👁 Image 👁 Image 0001 0011 👁 Image	5 ones 👁 Image 👁 Image 👁 Image 0011 0111 👁 Image 👁 Image 👁 Image 👁 Image 0101 1110 👁 Image 👁 Image 👁 Image 👁 Image 1001 0111 👁 Image
4 ones 👁 Image 👁 Image 👁 Image 0001 0111 👁 Image 👁 Image 👁 Image 👁 Image 0010 0111 👁 Image 👁 Image 👁 Image 👁 Image 0011 0011 👁 Image 👁 Image 👁 Image 👁 Image 0011 0110 👁 Image 👁 Image 👁 Image 👁 Image 0101 1010 👁 Image 👁 Image 👁 Image 👁 Image 0110 1001 👁 Image

A proposition is uniquely determined by the set of all cases, in which it is true.
This set could be called the proposition's validity set.
Two propositions are equal, when they have the same validity set.

The validity set of a negation is the complement of the initial proposition's validity set.
So to know the negation of a proposition, one has to know the set of all possible cases.

The set of all possible cases is the validity set of the tautology. It may be denoted 👁 {\displaystyle ~\Omega }
.
The empty set is the validity set of the contradiction.

Cases that can be the case and propositions that can be said are essentially different objects.
(Similar to outcomes and events in probability theory.)
When there are n possible cases, there are 2ⁿ possible propositions.
Among them are n elementary propositions (minterms). They have a 1-element validity set, and thus they are true in exactly one case.
(Cases and corresponding elementary propositions are easily mixed up - like outcomes and elementary events in probability theory.)

One may be interested, if an employee is a driver or a medic.
There are exactly four possibilities (cases), how he can have these qualities or not:

• 👁 Image
  He is neither D, nor M.
• 👁 Image
  He is D, but not M.
• 👁 Image
  He is not D, but M.
• 👁 Image
  He is D and M.

Exactly one of these statements will be true about a certain employee.
In respect to these qualities there are 2⁴ = 16 statements (propositions), one can say about this employee:

• 👁 Image
  "He is not."   (i.e.: False)
• 👁 Image
  "He is neither D, nor M."
• 👁 Image
  "He is D, but not M."
• 👁 Image
  "He is not M."

• 👁 Image
  "He is not D, but M."
• 👁 Image
  "He is not D."
• 👁 Image
  "He is either D or M."
• 👁 Image
  "He is not both D and M."

• 👁 Image
  "He is D and M."
• 👁 Image
  "He is either both D and M, or neither."
• 👁 Image
  "He is D."
• 👁 Image
  "If he is M, then he is also D."

• 👁 Image
  "He is M."
• 👁 Image
  "If he is D, then he is also M."
• 👁 Image
  "He is D or M."   (E.g.: "He must be D or M, otherwise he would not be part of this project.")
• 👁 Image
  "He is."   (i.e. True, the tautological case)

One may be interested, which of the following statements are true about a certain group of employees:

• 👁 Image
  "Someone is neither D, nor M."
• 👁 Image
  "Someone is D, but not M."
• 👁 Image
  "Someone is not D, but M."
• 👁 Image
  "Someone is D and M."

These statements don't contradict each other. At least one will be true about a certain group of employees.
(Assumed, that the group consists of at least one employee.)
There are 15 possible cases:

• 👁 Image
  All are neither D, nor M.
• 👁 Image
  All are D, but not M.
• 👁 Image
  There are D and not-D, which are all not M.

• 👁 Image
  All are not D, but M.
• 👁 Image
  All are not D, among them are M and not-M.
• 👁 Image
  There are D and M, but no one is both.
• 👁 Image
  No one is both D and M.

• 👁 Image
  All are D and M.
• 👁 Image
  All are both D and M, or neither.
• 👁 Image
  All are D.
• 👁 Image
  Those who are D are also M.

• 👁 Image
  All are M.
• 👁 Image
  Those who are M are also D.
• 👁 Image
  All are D or M.
• 👁 Image
  

So there are 2¹⁵ = 32768 propositions one can say about a particular group of employees:

• 👁 Image
  the tautology
  

with the validity set:

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

• 👁 Image
  "All D are M."
  

with the validity set:

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

• 👁 Image
  "No one is D and M."
  

with the validity set:

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

• 👁 Image
  "No one is D."
  

with the validity set:

👁 Image

👁 Image

👁 Image

• 👁 Image
  "All are D or M."
  

with the validity set:

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

👁 Image

• 👁 Image
  "All are D and M."
  

with the validity set:

👁 Image

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