consequence operator determined by a class of subsets

Theorem 1.

Let L be a set and let K be a subset of P(L). The the mapping C:P(L)P(L) defined as C(X)={YKXY} is a consequence operator.


We need to check that C satisfies the defining properties.

Property 1: Since every element of the set {YKXY} contains X, we have XC(X).

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


hence C(C(X))C(X). By the first property proven above, C(X)C(C(X)) so C(C(X))=C(X). Thus, CC=C.

Property 3: Let X and Y be two subsets of L such that XY. Then if, for some other subset Z of L, we have YZ, it follows that XZ. Hence,


so C(X)C(Y).

Title consequence operator determined by a class of subsets
Canonical name ConsequenceOperatorDeterminedByAClassOfSubsets
Date of creation 2013-03-22 16:29:45
Last modified on 2013-03-22 16:29:45
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type TheoremMathworldPlanetmath
Classification msc 03G25
Classification msc 03G10
Classification msc 03B22