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8.6.1 Containments ⊆ and intersections ⋂ of images and preimages 9 Sequences and collections of families of sets Toggle Sequences and collections of families of sets subsection
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
E is countably rectifiable when E equals the union of a countable family of rectifiable sets. E is countably ( ϕ , m ) {\displaystyle (\phi ,m)} rectifiable when ϕ {\displaystyle \phi } is a measure on X and there is a countably m {\displaystyle m} rectifiable set F such that ϕ ( E ∖ F ) = 0 {\displaystyle \phi (E\setminus F)=0} .
In set theory, the intersection of two sets and , denoted by , [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to . [2] Notation and terminology
Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
This set is denoted by ω 1 (or sometimes Ω). The set ω 1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 is distinct from ℵ 0. The definition of ℵ 1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ 0 and ℵ 1.