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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 }
 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.

File history

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Date/TimeThumbnailDimensionsUserComment
current22:19, 28 September 2024👁 Thumbnail for version as of 22:19, 28 September 2024
380 × 280 (351 bytes)WatchduckShade of red and thinner lines match other image sets.
15:32, 16 July 2024👁 Thumbnail for version as of 15:32, 16 July 2024
400 × 300 (614 bytes)AntonsusiValid SVG
23:10, 1 March 2024👁 Thumbnail for version as of 23:10, 1 March 2024
380 × 280 (351 bytes)Watchduckcleaner code and lighter red (overwritten with Pywikibot)
21:48, 24 February 2023👁 Thumbnail for version as of 21:48, 24 February 2023
380 × 280 (352 bytes)JoKalliauerrecreated, human redable, more symmetric
22:39, 19 November 2022👁 Thumbnail for version as of 22:39, 19 November 2022
512 × 373 (491 bytes)TSamuelCareful recompression via SVGOMG & vecta.io/nano, & verified via SVGCheck
14:10, 26 July 2009👁 Thumbnail for version as of 14:10, 26 July 2009
384 × 280 (3 KB)Watchduck
13:29, 26 January 2008👁 Thumbnail for version as of 13:29, 26 January 2008
615 × 463 (4 KB)Watchduck{{Information |Description= |Source=eigene arbeit |Date= |Author= Tilman Piesk |Permission= |other_versions= }}
16:03, 22 January 2008👁 Thumbnail for version as of 16:03, 22 January 2008
615 × 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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