File:Relation1111.svg

PLATO

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):

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

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

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

∅c

A = A

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

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

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

¬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

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

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

∅

A ${\displaystyle \scriptstyle \land }$ B

A = Ac

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

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.

Important relations

Set theory: Logic:	subset implication	disjoint contrary	subdisjoint subcontrary	equal equivalent	complementary contradictory

קובץ זה נטול זכויות יוצרים ולפיכך ברשות הציבור. יצירה מורכבת כולה, מיֶדַע שהוא נחלת הכלל, ואין לה מרכיב יצירתי מקורי.