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The circles in this Venn diagram can represent sets in set theory, or statements in propositional logic.

• In set theory it tells, that the left set is empty.
• In propositional logic it tells, that the left statement is never true.

In both interpretations 👁 Image
is the same as 👁 Image
👁 {\displaystyle ~\land ~}
👁 Image
.

👁 Image	👁 Image	👁 Image	👁 Image	👁 Image
	subset implication	disjoint contrary	subdisjoint subcontrary	equal equivalent	complementary contradictory

c

A = A

👁 1111

👁 1111

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

👁 1110

👁 0111

👁 1110

👁 0111

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

👁 1101

👁 0110

👁 1011

👁 1101

👁 0110

👁 1011

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

👁 1100

👁 0101

👁 1010

👁 0011

👁 1100

👁 0101

👁 1010

👁 0011

¬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

👁 0100

👁 1001

👁 0010

👁 0100

👁 1001

👁 0010

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

👁 1000

👁 0001

👁 1000

👁 0001

¬A 👁 {\displaystyle \scriptstyle \land }  ¬B

A 👁 {\displaystyle \scriptstyle \land }  B

A = Ac

👁 0000

👁 0000

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

👁 Image 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.   👁 Image 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. 👁 Image 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.) 👁 Image 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.

Public domainPublic domainfalsefalse

👁 Image

This work is ineligible for copyright and therefore in the public domain because it consists entirely of information that is common property and contains no original authorship.