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This Venn diagram is meant to represent arelationbetween

• two sets inset theory,
• or two statements inpropositional logicrespectively.

The relationtells, that the setisempty:=

It can be written as${\displaystyle A\subseteq B^{c}}$or as${\displaystyle B\subseteq A^{c}}$.
It tells, that the sets${\displaystyle ~A}$and${\displaystyle ~B}$have no elements in common:${\displaystyle A\cap B=\emptyset }$

Example: The set of positive numbers and the set of negative numbers are disjoint: No number is both positive and negative.
But they are not complementary sets, because the zero is neither positive nor negative.

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

Venn diagrams	written formulas
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle A\cap B=\emptyset }$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle B=B\setminus A}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle A=A\setminus B}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle A\cup B=A\Delta B}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle (A\Delta B)^{c}=(A\cup B)^{c}}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle A^{c}\cup B=A^{c}}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle A\cup B^{c}=B^{c}}$
${\displaystyle \Leftrightarrow }$=	${\displaystyle A\subseteq B^{c}}$${\displaystyle \Leftrightarrow }$${\displaystyle \emptyset ^{c}=(A\cap B)^{c}}$

The sign${\displaystyle \Leftrightarrow }$tells, that twostatements about setsmean the same.
The sign = tells, that twosetscontain the same elements.

The relationtells, that the statementis never true:${\displaystyle \Leftrightarrow }$

It can be written as${\displaystyle A\Rightarrow \neg B}$or as${\displaystyle B\Rightarrow \neg A}$.
It tells, that the statements${\displaystyle ~A}$and${\displaystyle ~B}$are never true together:${\displaystyle A\land B=false}$

Example: The statements"Number x is positive."and"Number x is negative."are contrary:
They can not be true together. But they are not contradictory, because both statements are false for x=0.

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

Venn diagrams	written formulas
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\land B\Leftrightarrow false}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle B\Leftrightarrow \neg A\land B}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\Leftrightarrow A\land \neg B}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\lor B\Leftrightarrow A\oplus B}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\leftrightarrow B\Leftrightarrow \neg (A\lor B)}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\rightarrow B\Leftrightarrow \neg A}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle A\leftarrow B\Leftrightarrow \neg B}$
${\displaystyle \equiv }$${\displaystyle \Leftrightarrow }$	${\displaystyle A\Rightarrow \neg B}$${\displaystyle \equiv }$${\displaystyle true\Leftrightarrow \neg (A\land B)}$

The sign${\displaystyle \equiv }$tells, that twostatements about statements about whatever objectsmean the same.
The sign${\displaystyle \Leftrightarrow }$tells, that twostatements about whatever objectsmean the same.

Important relations

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

∅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 tothis Hasse diagram.) In set theory the Venn diagrams represent the set, which is marked in red. These 15 relations, except the empty one, aremintermsand can be the case. The relations in the files below aredisjunctions.The red fields of their 4x4 matrices tell, in which ofthesecases 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. (Theexistential quantificationsfor the red intersections are combined byor. They can becombined by theexclusive oras well.) Relations likesubset 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.

This work isineligible forcopyrightand therefore in thepublic domainbecause it consists entirely of information that iscommon property and contains no original authorship.