It can be written as or as .
It tells, that the sets and have no elements in common:
Example: The set of positive numbers and the set of negative numbers are disjoint: No number is both positive and negative. But they are not complementary sets, because the zero is neither positive nor negative.
Under this condition several set operations, not equivalent in general, produce equivalent results.
These equivalences define disjoint sets:
Venn diagrams
written formulas
=
=
=
=
=
=
=
=
The sign tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.
The relation tells, that the statement is never true:
It can be written as or as .
It tells, that the statements and are never true together:
Example: The statements "Number x is positive." and "Number x is negative." are contrary: They can not be true together. But they are not contradictory, because both statements are false for x=0.
Under this condition several logic operations, not equivalent in general, produce equivalent results.
These equivalences define contrary statements:
Venn diagrams
written formulas
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.
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.)
{{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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