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Summary

One of 16 Venn diagrams, representing 2-ary Boolean functions like set operations and logical connectives:


Operations and relations in set theory and logic


∅c
A = A
1111 1111

Ac ∪ {\displaystyle \scriptstyle \cup } {\displaystyle \scriptstyle \cup } Bc true
A ↔ A
A ∪ {\displaystyle \scriptstyle \cup } {\displaystyle \scriptstyle \cup } B
A ⊆ {\displaystyle \scriptstyle \subseteq } {\displaystyle \scriptstyle \subseteq } Bc A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }A

A ⊇ {\displaystyle \scriptstyle \supseteq } {\displaystyle \scriptstyle \supseteq } Bc
1110 0111 1110 0111

A ∪ {\displaystyle \scriptstyle \cup } {\displaystyle \scriptstyle \cup } Bc ¬A ∨ {\displaystyle \scriptstyle \lor } {\displaystyle \scriptstyle \lor } ¬B
A → ¬B
A Δ {\displaystyle \scriptstyle \Delta } {\displaystyle \scriptstyle \Delta } B A ∨ {\displaystyle \scriptstyle \lor } {\displaystyle \scriptstyle \lor } B
A ← ¬B
Ac ∪ {\displaystyle \scriptstyle \cup } {\displaystyle \scriptstyle \cup } B
A ⊇ {\displaystyle \scriptstyle \supseteq } {\displaystyle \scriptstyle \supseteq } B A ⇒ {\displaystyle \scriptstyle \Rightarrow } {\displaystyle \scriptstyle \Rightarrow }¬B

A = Bc A ⇐ {\displaystyle \scriptstyle \Leftarrow } {\displaystyle \scriptstyle \Leftarrow }¬B

A ⊆ {\displaystyle \scriptstyle \subseteq } {\displaystyle \scriptstyle \subseteq } B
1101 0110 1011 1101 0110 1011

Bc A ∨ {\displaystyle \scriptstyle \lor } {\displaystyle \scriptstyle \lor } ¬B
A ← B
A A ⊕ {\displaystyle \scriptstyle \oplus } {\displaystyle \scriptstyle \oplus } B
A ↔ ¬B
Ac ¬A ∨ {\displaystyle \scriptstyle \lor } {\displaystyle \scriptstyle \lor } B
A → B
B
B = ∅ A ⇐ {\displaystyle \scriptstyle \Leftarrow } {\displaystyle \scriptstyle \Leftarrow }B

A = ∅c A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }¬B

A = ∅ A ⇒ {\displaystyle \scriptstyle \Rightarrow } {\displaystyle \scriptstyle \Rightarrow }B

B = ∅c
1100 0101 1010 0011 1100 0101 1010 0011
¬B

A ∩ {\displaystyle \scriptstyle \cap } {\displaystyle \scriptstyle \cap } Bc A

(A Δ {\displaystyle \scriptstyle \Delta } {\displaystyle \scriptstyle \Delta } B)c ¬A

Ac ∩ {\displaystyle \scriptstyle \cap } {\displaystyle \scriptstyle \cap } B B
B ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }false
A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }true

A = B A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }false
B ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }true
0100 1001 0010 0100 1001 0010
A ∧ {\displaystyle \scriptstyle \land } {\displaystyle \scriptstyle \land } ¬B

Ac ∩ {\displaystyle \scriptstyle \cap } {\displaystyle \scriptstyle \cap } Bc A ↔ {\displaystyle \scriptstyle \leftrightarrow } {\displaystyle \scriptstyle \leftrightarrow } B

A ∩ {\displaystyle \scriptstyle \cap } {\displaystyle \scriptstyle \cap } B ¬A ∧ {\displaystyle \scriptstyle \land } {\displaystyle \scriptstyle \land } B
A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\displaystyle \scriptstyle \Leftrightarrow }B
1000 0001 1000 0001
¬A ∧ {\displaystyle \scriptstyle \land } {\displaystyle \scriptstyle \land } ¬B

∅ A ∧ {\displaystyle \scriptstyle \land } {\displaystyle \scriptstyle \land } B

A = Ac
0000 0000
false
A ↔ ¬A A ⇔ {\displaystyle \scriptstyle \Leftrightarrow } {\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.



Public domainPublic domainfalsefalse
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.

Captions

Add a one-line explanation of what this file represents
Ein Venn-Diagram einer Kontravalenz. Da beide Kreise ausgefärbt sind (ein logisches Wahr, positiv, eine Eins, usw. darstellend), ist die Kontravalenz von beiden Kreisen zusammen eine unausgefärbte Form (logisches Falsch, negativ, Null darstellend).
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image/svg+xml

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Date/TimeThumbnailDimensionsUserComment
current21:06, 23 January 2025 Thumbnail for version as of 21:06, 23 January 2025 384 ×ばつ 280 (3 KB)Watchduck correct size
22:18, 28 September 2024 Thumbnail for version as of 22:18, 28 September 2024 410 ×ばつ 299 (3 KB)Watchduck Shade of red and thinner lines match other image sets.
15:29, 16 July 2024 Thumbnail for version as of 15:29, 16 July 2024 400 ×ばつ 300 (617 bytes)Antonsusi Valid SVG
23:14, 1 March 2024 Thumbnail for version as of 23:14, 1 March 2024 384 ×ばつ 280 (3 KB)Watchduck cleaner code and lighter red (overwritten with Pywikibot)
14:09, 26 July 2009 Thumbnail for version as of 14:09, 26 July 2009 384 ×ばつ 280 (3 KB)Watchduck
13:28, 26 January 2008 Thumbnail for version as of 13:28, 26 January 2008 615 ×ばつ 463 (4 KB)Watchduck {{Information |Description= |Source=eigene arbeit |Date= |Author= Tilman Piesk |Permission= |other_versions= }}
16:02, 22 January 2008 Thumbnail for version as of 16:02, 22 January 2008 615 ×ばつ 463 (4 KB)Watchduck {{Information |Description=Venn diagrams (sometimes called Johnston diagrams) concerning propositional calculus and set theory |Source=own work |Date=2008/Jan/22 |Author=Tilman Piesk |Permission=publich domain |other_versions= }}

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