Datei:Venn1100.svg

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Datei:Venn1100.svg
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Beschreibung

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.



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