The inclusion-exclusion principle for finitely many isolated sets

Journal of Symbolic Logic 51 (2):435-447 (1986)
Abstract
A nonnegative interger is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of $\alpha, \subset$ for inclusion and $\subset_+$ for proper inclusion. Let α, β 1 ,...,β k be subsets of some set ρ. Then α' stands for ρ-α and β 1 ⋯ β k for β 1 ∩ ⋯ ∩ β k . For subsets α 1 ,..., α r of ρ we write: $\alpha_\sigma - \{x \in v \ (\nabla i) \lbrack i \in \sigma \Rightarrow x \in \alpha_i\rbrack\} \text{for} \sigma \subset (1, \ldots, r),\\ s_i = \sum \{N(\alpha_\sigma) \mid \sigma \subset (1,\ldots, r) \& N(\sigma) = i\}, \text{for} 0 \leqq i \leqq r$ . Note that α 0 = v, hence s 0 = N(v). If the set v is finite, the classical inclusion-exclusion principle (abbreviated IEP) states $(a) N(\alpha_1 \cup \cdots \cup \alpha_r) = \sum^r_{t=1} (-1)^{t-1} s_t = \sum_{o \subset_+\sigma \subset (1,\ldots,r)}$ (b) N(α' 1 ⋯ α' r ) = ∑ r t=0 (-1) t s t = ∑ (-1) N(σ) N (α σ ). In this paper we generalize (a) and (b) to the case where α 1 ,⋯, α r are subsets of some countable but isolated set v. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalization of (a) and (b) are proved in § 3. Since they involve recursive distinctness, this notion is discussed in § 2. In § 4l we consider a natural extension of "the sum of the elements of a finite set σ" to the case where σ is countable. § 5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection λ of all isols which permit us to further generalize IEP by substituting μ(α) for $\operatorname{Req} \alpha$
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