Relatively Complemented Lattice

• If A and B are sets with

$A\backslash subseteq\; B$

then the complement of A relative to B (the interval involved is from the empty set to B) is

$B\backslash setminus\; A=\backslash left\backslash \{\backslash ,x\backslash in\; B\; :\; x\backslash not\backslash in\; A\backslash ,\backslash right\backslash \}.$

• If the lattice is a Boolean algebra, then the complement of b relative to the interval c is a ∨ (~ b) ∧ c. (In general, the expression x ∨ y ∧ z is ambiguous in Boolean algebra. But the fact that a ≤ b c removes the ambiguity in this case.) In the usual interpretation of Boolean algebra as a model of propositional logic, if a is a sufficient condition for b and c is a necessary condition for b, the complement of b relative to the interval c is the unique (up to logical equivalence) proposition d such that

• a is sufficient for d and c is necessary for d, and

• d becomes equivalent to b if one learns that a is false and c is true.

<< Previous Word Browser Next >>
english rugby union teams christian marclay mass fraction bobby jordan siglufjrur willemstad, netherlands antilles second rzhev sychevka offensive jaeger lecoultre chn castle caper gherkin	animal liberation sline bambi, a life in the woods phelsuma comorensis warp spasm river piddle cherem leicester tigers acrylonitrile nicolae paulescu heyting algebra	plain people lale andersen climbing shoe rochester international airport codasyl the allisons macmillan co. v. king greater rochester international airport c buildering tarifa	lcs magnum symbolic interactionism horgr kool g rap sergei golubitsky john barnes (author) list of people by name: prefixed van list of people by name: ve vh wigner's classification list of people by name: vi vk