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The indicator or characteristic function of a subset A of some set X maps elements of X to the codomain {,}. This mapping is surjective only when A is a non-empty proper subset of X . If A = X , {\displaystyle A=X,} then 1 A ≡ 1. {\displaystyle \mathbf {1} _{A}\equiv 1.}
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
The set of all cofinite subsets of (meaning those sets whose complement in is finite) is proper if and only if is infinite (or equivalently, is infinite), in which case is a filter on known as the Fréchet filter or the cofinite filter on . [10] [25] If is finite then is equal to the dual ideal ℘ (), which is not a filter.
Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, A ⊆ B and B ⊆ A is equivalent to A = B. [30] [8] The empty set is a subset of every set: ∅ ⊆ A. [17] If A is a subset of B, but A may not equal to B, then A is called a proper subset of B. This can be written A ...
A σ-algebra of subsets is a set algebra of subsets; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition. [ 2 ] The main use of σ-algebras is in the definition of measures ; specifically, the collection of those subsets for which a given measure is defined is ...
1. The difference of two sets: x~y is the set of elements of x not in y. 2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A product of sets / A quotient of a set by an equivalence ...
Subset is denoted by , proper subset by . The symbol ⊂ {\displaystyle \subset } may be used if the meaning is clear from context, or if it is not important whether it is interpreted as subset or as proper subset (for example, A ⊂ B {\displaystyle A\subset B} might be given as the hypothesis of a theorem whose conclusion is obviously true in ...