2010_lecFeb26

2010_lecFeb26 - CS245 Winter 2010 Lecture of Feb 26th Shai...

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Unformatted text preview: CS245 - Winter 2010, Lecture of Feb. 26th Shai Ben-David In this lecture we discuss definability for first order logic (FOL). There two different notions of definability. 0.1 Definability of a set of structures Definition Let L be a language. A set of L-structure, K is (strongly) definable if there exists a FOL formula in this language α such that K = { M : M | = α } . In that case, we say that α defines K . Examples: 1. Let L = ∅ . (a) K ≤ 2 def = { M : | U M | ≤ 2 } (where U M denotes the universe of the structure M ) is definable by the formula ∀ x ∀ y ∀ z ( x = y ∨ x = z ∨ y = z ). (b) K ≥ 2 def = { M : | U M | ≥ 2 } is definable by the formula ∃ x ∃ y ( x 6 = y ). 2. Let L = h R ( , ) i where R is a two-place relation symbol. (a) Let K equiv def = { M : R M is an equivalence relation } (recall that an equivalence relation is a relation that is symmetric, transitive and reflexive). Then K equiv is defined by α ≡ ( β 1 ∧ β 2 ∧ β 3 ), where • β...
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This note was uploaded on 04/07/2010 for the course CS 245 taught by Professor A during the Spring '08 term at Waterloo.

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2010_lecFeb26 - CS245 Winter 2010 Lecture of Feb 26th Shai...

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