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# Archivo:Relation1001.svg

## Resumen

This Venn diagram is meant to represent a relation between

### Set theory: The equivalence of sets

Two sets ${\displaystyle ~A}$ and ${\displaystyle ~B}$ are equivalent - i.e. contain the same elements - when all elements of ${\displaystyle ~A}$ are in ${\displaystyle ~B}$, and all elements of ${\displaystyle ~B}$ are in ${\displaystyle ~A}$.
In other words: If their symmetric difference is empty.

 ${\displaystyle \Leftrightarrow }$ ${\displaystyle \land }$ ${\displaystyle \Leftrightarrow }$ = ${\displaystyle ~A=B~}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle A\subseteq B}$ ${\displaystyle \land }$ ${\displaystyle B\subseteq A}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Delta B}$ = ${\displaystyle ~\emptyset }$

Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent sets:

 ${\displaystyle \Leftrightarrow }$ = = = ${\displaystyle ~A=B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Delta B}$ = ${\displaystyle ~A\setminus B}$ = ${\displaystyle ~B\setminus A}$ = ${\displaystyle \emptyset }$

 ${\displaystyle \Leftrightarrow }$ = = = ${\displaystyle ~A=B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\cup B}$ = ${\displaystyle ~A}$ = ${\displaystyle ~B}$ = ${\displaystyle ~A\cap B}$

 ${\displaystyle \Leftrightarrow }$ = = = ${\displaystyle ~A=B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A^{c}\cup B^{c}}$ = ${\displaystyle ~B^{c}}$ = ${\displaystyle ~A^{c}}$ = ${\displaystyle ~A^{c}\cap B^{c}}$

 ${\displaystyle \Leftrightarrow }$ = = = ${\displaystyle ~A=B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~\emptyset ^{c}}$ = ${\displaystyle ~A\cup B^{c}}$ = ${\displaystyle ~A^{c}\cup B}$ = ${\displaystyle ~(A\Delta B)^{c}}$

The sign ${\displaystyle \Leftrightarrow }$ tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.

### Propositional logic: The equivalence of statements

Two statements ${\displaystyle ~A}$ and ${\displaystyle ~B}$ are equivalent - i.e. together true or together false - when ${\displaystyle ~A}$ implies ${\displaystyle ~B}$, and ${\displaystyle ~B}$ implies ${\displaystyle ~A}$.
In other words: If their exclusive or is never true.

 ${\displaystyle \equiv }$ ${\displaystyle \land }$ ${\displaystyle \equiv }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Leftrightarrow B~}$ ${\displaystyle \equiv }$ ${\displaystyle A\Rightarrow B}$ ${\displaystyle \land }$ ${\displaystyle B\Rightarrow A}$ ${\displaystyle \equiv }$ ${\displaystyle ~A\oplus B}$ ${\displaystyle ~\Leftrightarrow }$ ${\displaystyle ~false}$

Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent statements:

 ${\displaystyle \equiv }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Leftrightarrow B}$ ${\displaystyle \equiv }$ ${\displaystyle ~A\oplus B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\land \neg B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg A\land B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~false}$

 ${\displaystyle \equiv }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Leftrightarrow B}$ ${\displaystyle \equiv }$ ${\displaystyle ~A\lor B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\land B}$

 ${\displaystyle \equiv }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Leftrightarrow B}$ ${\displaystyle \equiv }$ ${\displaystyle ~\neg A\lor \neg B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~\neg B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~\neg A}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~\neg A\land \neg B}$

 ${\displaystyle \equiv }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\Leftrightarrow B}$ ${\displaystyle \equiv }$ ${\displaystyle ~true}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\lor \neg B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg A\lor B}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~A\leftrightarrow B}$

Especially the last line is important:
The logical equivalence ${\displaystyle A\Leftrightarrow B}$ tells, that the material equivalence ${\displaystyle A\leftrightarrow B}$ is always true.
The material equivalence ${\displaystyle A\leftrightarrow B}$ is the same as ${\displaystyle \neg (A\oplus B)}$, the negated exclusive or.
Note: Names like logical equivalence and material equivalence are used in many different ways, and shouldn't be taken too serious.

The sign ${\displaystyle \equiv }$ tells, that two statements about statements about whatever objects mean the same.
The sign ${\displaystyle \Leftrightarrow }$ tells, that two statements about whatever objects mean the same.

 Set theory: subset disjoint subdisjoint equal complementary Logic: implication contrary subcontrary equivalent contradictory

## Operations and relations in set theory and logic

 ∅c A = A Ac ${\displaystyle \scriptstyle \cup }$ Bc trueA ↔ A A ${\displaystyle \scriptstyle \cup }$ B A ${\displaystyle \scriptstyle \subseteq }$ Bc A${\displaystyle \scriptstyle \Leftrightarrow }$A A ${\displaystyle \scriptstyle \supseteq }$ Bc A ${\displaystyle \scriptstyle \cup }$ Bc ¬A ${\displaystyle \scriptstyle \lor }$ ¬BA → ¬B A ${\displaystyle \scriptstyle \Delta }$ B A ${\displaystyle \scriptstyle \lor }$ BA ← ¬B Ac ${\displaystyle \scriptstyle \cup }$ B A ${\displaystyle \scriptstyle \supseteq }$ B A${\displaystyle \scriptstyle \Rightarrow }$¬B A = Bc A${\displaystyle \scriptstyle \Leftarrow }$¬B A ${\displaystyle \scriptstyle \subseteq }$ B Bc A ${\displaystyle \scriptstyle \lor }$ ¬BA ← B A A ${\displaystyle \scriptstyle \oplus }$ BA ↔ ¬B Ac ¬A ${\displaystyle \scriptstyle \lor }$ BA → B B B = ∅ A${\displaystyle \scriptstyle \Leftarrow }$B A = ∅c A${\displaystyle \scriptstyle \Leftrightarrow }$¬B A = ∅ A${\displaystyle \scriptstyle \Rightarrow }$B B = ∅c ¬B A ${\displaystyle \scriptstyle \cap }$ Bc A (A ${\displaystyle \scriptstyle \Delta }$ B)c ¬A Ac ${\displaystyle \scriptstyle \cap }$ B B B${\displaystyle \scriptstyle \Leftrightarrow }$false A${\displaystyle \scriptstyle \Leftrightarrow }$true A = B A${\displaystyle \scriptstyle \Leftrightarrow }$false B${\displaystyle \scriptstyle \Leftrightarrow }$true A ${\displaystyle \scriptstyle \land }$ ¬B Ac ${\displaystyle \scriptstyle \cap }$ Bc A ${\displaystyle \scriptstyle \leftrightarrow }$ B A ${\displaystyle \scriptstyle \cap }$ B ¬A ${\displaystyle \scriptstyle \land }$ B A${\displaystyle \scriptstyle \Leftrightarrow }$B ¬A ${\displaystyle \scriptstyle \land }$ ¬B ∅ A ${\displaystyle \scriptstyle \land }$ B A = Ac falseA ↔ ¬A A${\displaystyle \scriptstyle \Leftrightarrow }$¬A These sets (statements) have complements (negations).They are in the opposite position within this matrix. These relations are statements, and have negations.They are shown in a separate matrix in the box below.

 Esta imagen no es elegible para estar sujeta a derecho de autor y por tanto está en el dominio público, porque consiste enteramente en información que es de propiedad común y carece de autoría original.
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