User:Watchduck/Logic

Operations

Binary logical connectives       (zoom in)       (compare)

Relations

Minterm relations (relations that can be the case) (zoom in)

Maxterm relations (zoom in)

Negative statements combined by AND (zoom in)

Affirmative statements combined by OR (zoom in)

XNOR and XOR
Negative statements combined by XNOR (zoom in) Affirmative statements combined by XOR (zoom in)

Affirmative statements combined by AND (zoom in)

Negative statements combined by OR (zoom in)

XNOR and XOR
Affirmative statements combined by XNOR (zoom in) Negative statements combined by XOR (zoom in)

In different universes

Are operations relations in a 1-element universe?
Relations in a 1-element universe (The matrices in the center of this file ...) Operations (... are the same as in the bottom right corner of this file.)

1-element universe:

$~\land$

$~\Leftrightarrow$

2-element universe:

$~\land$

$~\Leftrightarrow$

3-element universe:

$~\land$

$~\Leftrightarrow$

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

$~\land$

$~\Leftrightarrow$

5-element universe:

(zoom in)

$~\land$

(zoom in)

$~\Leftrightarrow$

(zoom in)

Parity relations

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 where the others have a Sierpinski triangle .

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

$~\land$

$~\Leftrightarrow$

2-element universe:

$~\land$

$~\Leftrightarrow$

3-element universe:

$~\land$

$~\Leftrightarrow$

4-element universe:

$~\land$

$~\Leftrightarrow$

5-element universe:

(zoom in)

$~\land$

(zoom in)

$~\Leftrightarrow$

(zoom in)

Examples

In is an even number of elements.
: In is an odd number of elements. $\land ~$ : In is an odd number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an even number of elements. : In is an even number of elements. $\land ~$ : In is an even number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an even number of elements. $\lor ~$ $\Leftrightarrow ~$ In is an even number of elements.

In is an odd number of elements.
: In is an odd number of elements. $\land ~$ : In is an even number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an odd number of elements. : In is an even number of elements. $\land ~$ : In is an odd number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an odd number of elements. $\lor ~$ $\Leftrightarrow ~$ In is an odd number of elements.

In is an odd number of elements.

$\oplus ~$

In is an odd number of elements.

$\Leftrightarrow ~$

Either in or in is an odd number of elements.

This is a different way to write the same:

In is an odd number of elements.
: In is an odd number of elements. $\land ~$ : In is an even number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an odd number of elements. : In is an even number of elements. $\land ~$ : In is an odd number of elements. $\Leftrightarrow ~$ $\Rightarrow$ In is an odd number of elements. $\lor ~$ $\Leftrightarrow ~$ In is an odd number of elements.

This is what the exclusive or excludes:

In and in is an odd number of elements.

It's not to be confused with:

: In is an odd number of elements.

Just another example:

What's that?

3-ary relations

Venn and Euler diagrams

$A\subseteq B$	$\land ~$	$B\subseteq C$	$\Leftrightarrow$	$A\subseteq B\subseteq C$
1011 1011	$\land ~$	1100 1111	$\Leftrightarrow ~$	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 0000 0000 0000 0001	7 and 8 ones 0111 1111 1111 1111
2 ones 1000 0001 0000 0101 0010 0001	6 ones 0111 1011 0101 1111 0111 1110
3 ones 0001 0110 1000 0101 0001 0011	5 ones 0011 0111 0101 1110 1001 0111
4 ones 0001 0111 0010 0111 0011 0011 0011 0110 0101 1010 0110 1001

Propositional calculus examples: drivers and medics

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 $~\Omega$.
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.)

A single employee

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

• He is neither D, nor M.
• He is D, but not M.
• He is not D, but M.
• 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:

• 
• "He is neither D, nor M."
• "He is D, but not M."
• 

• "He is not D, but M."
• 
• "He is either D or M."
• 

• "He is D and M."
• 
• "He is D."
• 

• "He is M."
• 
• "He is D or M."   (E.g.: "He must be D or M, otherwise he would not be part of this project.")
• 

A group of employees

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

• "Someone is neither D, nor M."
• "Someone is D, but not M."
• "Someone is not D, but M."
• "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:

• All are neither D, nor M.
• All are D, but not M.
• There are D and not-D, which are all no M.

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

• All are D and M.
• 
• 
• 

• 
• 
• 
• 

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

Examples

• the tautology
  

with the validity set:

{

}

• "All D are M."
  

with the validity set:

{

}

• "No one is D and M."
  

with the validity set:

{

}

• "No one is D."
  

with the validity set:

{

}

• "All are D or M."
  

with the validity set:

{

}

• "All are D and M."
  

with the validity set:

{

}