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Captions

Captions

Used to compare 2 things. my putting 2 circles together.

PLATO

Summary

[edit]

This Venn diagram is meant to represent the special case of a relation between two sets in set theory,
or two statements in propositional logic respectively.


Example (for sets):

👁 Image
 👁 {\displaystyle ~=~}
 👁 Image
   ðŸ‘ {\displaystyle ~\Leftrightarrow ~}
   ðŸ‘ Image

The sets 👁 {\displaystyle ~A~}
 and 👁 {\displaystyle ~A\cap B~}
 are not equivalent in general: They are equivalent when 👁 {\displaystyle ~A\subseteq B~}
.


👁 Image
 👁 {\displaystyle ~=~}
 👁 Image
   ðŸ‘ {\displaystyle ~\Leftrightarrow ~}
   ðŸ‘ Image

The sets 👁 {\displaystyle ~A~}
 and 👁 {\displaystyle ~A~}
 are always equivalent: 👁 {\displaystyle ~A=A~}
 is a tautology.


This is what 👁 Image
 tells. It's shown at the top of the right one of the below diagrams.

 
c		  
A = A
👁 1111
 👁 1111
 
Ac ðŸ‘ {\displaystyle \scriptstyle \cup }
 Bc		 true
A ↔ A		  
A ðŸ‘ {\displaystyle \scriptstyle \cup }
 B		  
A ðŸ‘ {\displaystyle \scriptstyle \subseteq }
 Bc		 A👁 {\displaystyle \scriptstyle \Leftrightarrow }
A
 		  
A ðŸ‘ {\displaystyle \scriptstyle \supseteq }
 Bc
👁 1110
 👁 0111
 👁 1110
 👁 0111
 
A ðŸ‘ {\displaystyle \scriptstyle \cup }
 Bc		 ÂŽA ðŸ‘ {\displaystyle \scriptstyle \lor }
 ÂŽB
A → ÂŽB		  
A ðŸ‘ {\displaystyle \scriptstyle \Delta }
 B		 A ðŸ‘ {\displaystyle \scriptstyle \lor }
 B
A ← ÂŽB		  
Ac 👁 {\displaystyle \scriptstyle \cup }
 B		  
A 👁 {\displaystyle \scriptstyle \supseteq }
 B		 A👁 {\displaystyle \scriptstyle \Rightarrow }
ÂŽB
 		  
A = Bc		 A👁 {\displaystyle \scriptstyle \Leftarrow }
ÂŽB
 		  
A 👁 {\displaystyle \scriptstyle \subseteq }
B
👁 1101
 👁 0110
 👁 1011
 👁 1101
 👁 0110
 👁 1011
 
Bc		 A ðŸ‘ {\displaystyle \scriptstyle \lor }
 ÂŽB
A ← B		  
A		 A ðŸ‘ {\displaystyle \scriptstyle \oplus }
 B
A ↔ ÂŽB		  
Ac		 ÂŽA ðŸ‘ {\displaystyle \scriptstyle \lor }
 B
A → B		  
B		  
B = 		A👁 {\displaystyle \scriptstyle \Leftarrow }
B
 		  
A = c		 A👁 {\displaystyle \scriptstyle \Leftrightarrow }
ÂŽB
 		  
A = 		A👁 {\displaystyle \scriptstyle \Rightarrow }
B
 		  
B = c
👁 1100
 👁 0101
 👁 1010
 👁 0011
 👁 1100
 👁 0101
 👁 1010
 👁 0011
ÂŽB
 		  
A ðŸ‘ {\displaystyle \scriptstyle \cap }
 Bc		 A
 		  
(A ðŸ‘ {\displaystyle \scriptstyle \Delta }
 B)c		 ÂŽA
 		  
Ac ðŸ‘ {\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 ðŸ‘ {\displaystyle \scriptstyle \land }
 ÂŽB
 		  
Ac ðŸ‘ {\displaystyle \scriptstyle \cap }
 Bc		 A ðŸ‘ {\displaystyle \scriptstyle \leftrightarrow }
 B
 		  
A ðŸ‘ {\displaystyle \scriptstyle \cap }
 B		 ÂŽA ðŸ‘ {\displaystyle \scriptstyle \land }
 B
 		 A👁 {\displaystyle \scriptstyle \Leftrightarrow }
B
 
👁 1000
 👁 0001
 👁 1000
 👁 0001
ÂŽA ðŸ‘ {\displaystyle \scriptstyle \land }
 ÂŽB
 		  
 A ðŸ‘ {\displaystyle \scriptstyle \land }
 B
 		  
A = Ac
👁 0000
 👁 0000
false
A ↔ ÂŽA		 A👁 {\displaystyle \scriptstyle \Leftrightarrow }
ÂŽA
 
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.
 
 



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



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
current22:51, 7 May 2010👁 Thumbnail for version as of 22:51, 7 May 2010
384 × 280 (3 KB)Watchduck (talk | contribs)layout change
18:01, 26 July 2009👁 Thumbnail for version as of 18:01, 26 July 2009
384 × 280 (4 KB)Watchduck (talk | contribs)
16:16, 10 April 2009👁 Thumbnail for version as of 16:16, 10 April 2009
615 × 463 (4 KB)Watchduck (talk | contribs){{Information |Description={{en|1=Venn diagrams of the sixteen 2-ary Boolean '''relations'''. Black (0) marks empty areas (compare empty set). White (1) means, that there ''could'' be something. There are corresponding diagrams of th

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