20Summary of results
In Propositions 30 to 38, X and Y are subsets of a domain U.
Proposition 1 (double inclusion). For any sets X and Y, if X ⊂ Y and Y ⊂ X, then X = Y.
Proposition 2 (every set is a subset of itself). Every set is a subset of itself.
Proposition 3 (transitivity of inclusion). For any sets X, Y and Z, if X ⊂ Y and Y ⊂ Z, then X ⊂ Z.
Proposition 4 (an empty set is a subset of anything). For any sets X and Y, if X contains no elements, then X ⊂ Y.
Proposition 5 (the empty set is unique). Any two sets with no elements are identical.
Proposition 6 (the only subset of ∅). For any set X, if X ⊂ ∅, then X = ∅.
Proposition 7 (equivalent criteria give the same set). If two sets have equivalent inclusion criteria, then they are identical.
Proposition 8 (an intersection is a subset of each set). For any sets X and Y, X ∩ Y ⊂ X and X ∩ Y ⊂ Y.
Proposition 9 (intersection respects inclusion). For any sets X, Y and Z, if X ⊂ Y, then X ∩ Z ⊂ Y ∩ Z.
Proposition 10 (intersection commutes). For any sets X and Y, X ∩ Y = Y ∩ X.
Proposition 11 (intersection is associative). For any sets X, Y and Z, (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z).
Proposition 12 (intersecting a set with itself). For any set X, X ∩ X = X.
Proposition 13 (intersecting with the empty set). For any set X, X ∩ ∅ = ∅.
Proposition 14 (inclusion via intersection). For any sets X and Y, X ⊂ Y if and only if X ∩ Y = X.
Proposition 15 (each set is a subset of the union). For any sets X and Y, X ⊂ X ∪ Y and Y ⊂ X ∪ Y.
Proposition 16 (union commutes). For any sets X and Y, X ∪ Y = Y ∪ X.
Proposition 17 (union is associative). For any sets X, Y and Z, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z).
Proposition 18 (uniting a set with itself). For any set X, X ∪ X = X.
Proposition 19 (uniting with the empty set). For any set X, X ∪ ∅ = X.
Proposition 20 (union respects inclusion). For any sets X, Y and Z, if X ⊂ Y, then X ∪ Z ⊂ Y ∪ Z.
Proposition 21 (the union is the smallest set containing both). For any sets X, Y and Z, if X ⊂ Z and Y ⊂ Z, then X ∪ Y ⊂ Z.
Proposition 22 (inclusion via union). For any sets X and Y, X ⊂ Y if and only if X ∪ Y = Y.
Proposition 23 (intersection distributes over union). For any sets X, Y and Z, X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z).
Proposition 24 (union distributes over intersection). For any sets X, Y and Z, X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z).
Proposition 25 (absorption for union). For any sets X and Y, X ∪ (X ∩ Y) = X.
Proposition 26 (absorption for intersection). For any sets X and Y, X ∩ (X ∪ Y) = X.
Proposition 27 (a relative complement is a subset). For any sets X and Y, X ∖ Y ⊂ X.
Proposition 28 (subtracting the empty set). For any set X, X ∖ ∅ = X.
Proposition 29 (subtracting a set from itself). For any set X, X ∖ X = ∅.
Proposition 30 (a set and its complement exhaust the domain). X ∪ Xc = U.
Proposition 31 (a set and its complement are disjoint). X ∩ Xc = ∅.
Proposition 32 (the complement of a complement). (Xc)c = X.
Proposition 33 (the complement of the empty set). ∅c = U.
Proposition 34 (the complement of the domain). Uc = ∅.
Proposition 35 (relative complement via complement). X ∖ Y = X ∩ Yc.
Proposition 36 (De Morgan, for intersection). (X ∩ Y)c = Xc ∪ Yc.
Proposition 37 (De Morgan, for union). (X ∪ Y)c = Xc ∩ Yc.
Proposition 38 (complements reverse inclusion). X ⊂ Y if and only if Yc ⊂ Xc.
Proposition 39 (a set is disjoint from what is taken off another). For any sets X and Y, X and Y ∖ X are disjoint.
Proposition 40 (disjointness passes to subsets). For any sets X, Y and Z, if X and Y are disjoint and Z ⊂ X, then Z and Y are disjoint.
Proposition 41 (splitting a set around a subset). For any sets X and Y, if X ⊂ Y, then Y = X ∪ (Y ∖ X).
Proposition 42 (splitting a union into disjoint pieces). For any sets X and Y, X ∪ Y = X ∪ (Y ∖ X).
Proposition 43 (splitting a set by another set). For any sets X and Y, Y = (Y ∖ X) ∪ (X ∩ Y).
Proposition 44 (a union of sets disjoint from Z is disjoint from Z). For any sets X, Y and Z, if X and Z are disjoint and Y and Z are disjoint, then X ∪ Y and Z are disjoint.
Proposition 45 (each set is a subset of the union of its collection). For any collection ℱ and any X ∈ ℱ, X ⊂ ⋃ ℱ.
Proposition 46 (the intersection of a collection is a subset of each). For any collection ℱ and any X ∈ ℱ, ⋂ ℱ ⊂ X.
Proposition 47 (each element lies in exactly one cell). If ℱ is a partition of X, then every element of X belongs to exactly one element of ℱ.