User:Watchduck/Logic

Relations

In different universes

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.

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

 $\oplus ~$ $\Leftrightarrow ~$ This is a different way to write the same:

This is what the exclusive or excludes:

It's not to be confused with:

Just another example:

3-ary relations

 $A\subseteq B$ $\land ~$ $B\subseteq C$ $\Leftrightarrow$ $A\subseteq B\subseteq C$ $\land ~$ $\Leftrightarrow ~$ 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:

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:

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:

A group of employees

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

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:

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

Examples

 with the validity set: { }