# De Morgan's Laws

The **de-morgan's laws** (and often **de-Morganist rules** ) are two basic rules for logical statements. They were named after the mathematician Augustus De Morgan , although they were already known to the medieval logician Wilhelm von Ockham . They apply in all Boolean algebras . They are particularly important in propositional logic and set theory . In technology, they are important for creating interlocks and programs.

## Laws

In logic they are :

- not (a and b) is equivalent to ((not a) or (not b))
- not (a or b) is equivalent to ((not a) and (not b))

In mathematics, one can find numerous different representations of the De Morgan laws of propositional logic :

- or with a different notation:

The validity of *De Morgan's laws* can be proven using truth *tables* .

Its equivalent in set theory is (where *A is* the complement of *A* , the symbol for the intersection of two sets and the symbol for the union of two sets):

The rules can also be extended to link any number of elements. So for any finite, countable or non-countable index set *I* :

- and .

## Inferences

A conjunction (AND link) can be represented by three negations and a disjunction (NOT or OR links) with the help of De-Morgan's law :

Accordingly, a disjunction can be represented by three negations and a conjunction:

## application

The *de-morgan's laws* have important applications in discrete mathematics , the electrical , the physics and computer science . In particular, the de-morgan's laws are in the design of digital circuits used to the types of logic used switching elements replaced with another or to save components.

## Examples

### Example from everyday life

Suppose a person likes to drink coffee: To express that he only drinks it black and without sugar, he can make the following statements:

*If there is milk or sugar in it, then I will not drink the coffee.*

Converted from de Morgan and counterposition :

*When I drink the coffee, there is no milk or sugar in it.*

Both statements are of equal value.

### Example in set theory

It's supposed to be based on the relationship

the validity of De Morgan's rules are illustrated. There are two sets A and B, which are subsets of a superset Ω. The graph 1 shows the location of the quantities and its counter quantities A and B .

Figure 2 shows how it is formed. Figure 3 shows the complement to , and you can see that both sets are equal.

Division of the superset into A and B. |

*One interpretation would be:*

In an acceptance test, high-quality chef's knives are checked to see whether the cutting edge is free of defects (quantity A) and whether the cutting edge is properly anchored in the handle (quantity B). A knife is not accepted, if there is the amount A or the amount of B is one or two, that is, when at least one complaint is present: . The knife is accepted if it fulfills both requirements, that is, if it belongs to the crowd , that is, it is not accepted if it belongs to.