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This Venn diagram is meant to represent a relation between


Set theory: The equivalence of sets

Two sets and are equivalent - i.e. contain the same elements - when all elements of are in , and all elements of are in .
In other words: If their symmetric difference is empty.

Relation1001.svg          Relation1011.svg Relation1101.svg          Venn0110.svg = Venn0000.svg
                  =

Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent sets:


Relation1001.svg              Venn0110.svg     =     Venn0100.svg     =     Venn0010.svg     =     Venn0000.svg
= = =


Relation1001.svg              Venn0111.svg     =     Venn0101.svg     =     Venn0011.svg     =     Venn0001.svg
= = =


Relation1001.svg              Venn1110.svg     =     Venn1100.svg     =     Venn1010.svg     =     Venn1000.svg
= = =


Relation1001.svg              Venn1111.svg     =     Venn1101.svg     =     Venn1011.svg     =     Venn1001.svg
= = =

The sign tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.


Propositional logic: The equivalence of statements

Two statements and are equivalent - i.e. together true or together false - when implies , and implies .
In other words: If their exclusive or is never true.

Relation1001.svg          Relation1011.svg Relation1101.svg          Venn0110.svg Venn0000.svg
                 

Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent statements:


Relation1001.svg              Venn0110.svg          Venn0100.svg          Venn0010.svg          Venn0000.svg


Relation1001.svg              Venn0111.svg          Venn0101.svg          Venn0011.svg          Venn0001.svg


Relation1001.svg              Venn1110.svg          Venn1100.svg          Venn1010.svg          Venn1000.svg


Relation1001.svg              Venn1111.svg          Venn1101.svg          Venn1011.svg          Venn1001.svg

Especially the last line is important:
The logical equivalence tells, that the material equivalence is always true.
The material equivalence is the same as , the negated exclusive or.
Note: Names like logical equivalence and material equivalence are used in many different ways, and shouldn't be taken too serious.

The sign tells, that two statements about statements about whatever objects mean the same.
The sign tells, that two statements about whatever objects mean the same.




Important relations
Relation1011.svg Relation1110.svg Relation0111.svg Relation1001.svg Relation0110.svg
Set theory: subset disjoint subdisjoint equal complementary
Logic: implication contrary subcontrary equivalent contradictory


Operations and relations in set theory and logic

 
c
          
A = A
1111 1111
 
Ac  Bc
true
A ↔ A
 
 B
 
 Bc
AA
 
 
 Bc
1110 0111 1110 0111
 
 Bc
¬A  ¬B
A → ¬B
 
 B
 B
A ← ¬B
 
Ac B
 
A B
A¬B
 
 
A = Bc
A¬B
 
 
A  B
1101 0110 1011 1101 0110 1011
 
Bc
 ¬B
A ← B
 
A
 B
A ↔ ¬B
 
Ac
¬A  B
A → B
 
B
 
B =
AB
 
 
A = c
A¬B
 
 
A =
AB
 
 
B = c
1100 0101 1010 0011 1100 0101 1010 0011
¬B
 
 
 Bc
A
 
 
(A  B)c
¬A
 
 
Ac  B
B
 
Bfalse
 
Atrue
 
 
A = B
Afalse
 
Btrue
 
0100 1001 0010 0100 1001 0010
 ¬B
 
 
Ac  Bc
 B
 
 
 B
¬A  B
 
AB
 
1000 0001 1000 0001
¬A  ¬B
 
 
 B
 
 
A = Ac
0000 0000
false
A ↔ ¬A
A¬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.



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