PlanetMath: consequence operator determined by a class of subsets

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consequence operator determined by a class of subsets

(Theorem)

Theorem 1 Let be a set and let be a subset of $\mathcal{P}(L)$ . The the mapping $C \colon \mathcal{P}(L) \to \mathcal{P}(L)$ defined as $C(X) = \cap \{ Y \in K \mid X \subseteq Y\}$ is a consequence operator.

Proof. We need to check that

satisfies the defining properties.

Property 1: Since every element of the set $\{ Y \in K \mid X \subseteq Y \}$ contains , we have $X \subseteq C(X)$ .

Property 2: For every element of such that $X \subseteq Y$ , it also is the case that $C(X) \subseteq Y$ because an intersection of a family of sets is a subset of any member of the family. In other words (or rather, symbols),

$\displaystyle \{ Y \in K \mid X \subseteq Y \} \subseteq \{ Y \in K \mid C(X) \subseteq Y \},$

hence $C(C(X)) \subseteq C(X)$ . By the first property proven above, $C(X) \subseteq C(C(X))$ so

. Thus, $C \circ C = C$ .

Property 3: Let and be two subsets of such that $X \subseteq Y$ . Then if, for some other subset of , we have $Y \subset Z$ , it follows that $X \subset Z$ . Hence,

$\displaystyle \{ Z \in K \mid Y \subseteq Z \} \subseteq \{ Z \in K \mid X \subseteq Z \},$

so $C(X) \subseteq C(Y)$ .

$\qedsymbol$

"consequence operator determined by a class of subsets" is owned by rspuzio. [ full author list (2) ]

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Cross-references: intersection, contains, property, defining properties, satisfies, consequence operator, mapping, subset
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This is version 7 of consequence operator determined by a class of subsets, born on 2006-12-22, modified 2006-12-24.
Object id is 8671, canonical name is ConsequenceOperatorDeterminedByAClassOfSubsets.
Accessed 710 times total.

Classification:

AMS MSC:	03B22 (Mathematical logic and foundations :: General logic :: Abstract deductive systems)
	03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures)
	03G25 (Mathematical logic and foundations :: Algebraic logic :: Other algebras related to logic)

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