ファイル:Relation1001.svg

概要

This Venn diagram is meant to represent a relation between

• two sets in set theory,
• or two statements in propositional logic respectively.

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.

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Important relations

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

true A ↔ 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 ${\displaystyle \scriptstyle \lor }$ ¬B A → ¬B

A ${\displaystyle \scriptstyle \Delta }$ B

A ${\displaystyle \scriptstyle \lor }$ B A ← ¬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 ${\displaystyle \scriptstyle \lor }$ ¬B A ← B

A

A ${\displaystyle \scriptstyle \oplus }$ B A ↔ ¬B

Ac

¬A ${\displaystyle \scriptstyle \lor }$ B A → 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

false A ↔ ¬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.

more relations

The operations, arranged in the same matrix as above. The 2x2 matrices show the same information like the Venn diagrams. (This matrix is similar to this Hasse diagram.)    In set theory the Venn diagrams represent the set, which is marked in red.   These 15 relations, except the empty one, are minterms and can be the case. The relations in the files below are disjunctions. The red fields of their 4x4 matrices tell, in which of these cases the relation is true. (Inherently only conjunctions can be the case. Disjunctions are true in several cases.) In set theory the Venn diagrams tell, that there is an element in every red, and there is no element in any black intersection. Negations of the relations in the matrix on the right. In the Venn diagrams the negation exchanges black and red.   In set theory the Venn diagrams tell, that there is an element in one of the red intersections. (The existential quantifications for the red intersections are combined by or. They can be combined by the exclusive or as well.) Relations like subset and implication, arranged in the same kind of matrix as above.   In set theory the Venn diagrams tell, that there is no element in any black intersection.

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