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notes_22_01_08 - Applied Logic Lecture Notes The following...

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Applied Logic Lecture Notes 22-01-08 The following are brief lecture notes and some additional remarks on the Axiom of Choice. Much of the material can be found in Chapter VI of the text, but it is conve- niently gathered here for future reference. The problems are completely optional. 1 A Taxonomy of Binary Relations Here are some common properties of a binary relation R (with universe V ). Reflexivity. xRx Irreflexivity. xR x Symmetry. xRy yRx Antisymmetry. xRy yRx x = y Transitivity. xRy yRz xRz Trichotomy. xRy x = y yRx Dichotomy. xRy yRx By convention, x, y, z above are implicitly universally quantified, i.e. R is reflexive iff xRx for every x (in V ). Here are some common types of relations: Simple Graph. A symmetric, irreflexive relation. Equivalence Relation. A symmetric, reflexive, transitive relation. Strict Partial Order. An irreflexive, transitive relation. Strict Linear Order. A strict partial order that satisfies the trichotomy law. (Non-Strict) Partial Order. An antisymmetric, reflexive, transitive relation. (Non-Strict) Linear Order. A partial order that satisfies the dichotomy law. Quasi Order. A reflexive, transitive relation. The text defines “partial order” and “linear order” to mean “strict linear order” and “strict partial order,” respectively. This is not the usual convention, but the next proposition shows that the two notions are essentially interchangeable. Proposition. Let < and be relations with the same universe V such that x y x < y x = y and x< y x y x y . (So is definable from < , and vice versa.) Then: < is a strict partial order if and only if is a partial order 1
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< is a strict linear order if and only if is a linear order So every strict partial/linear order corresponds to a unique (non-strict) partial/linear order, and vice versa .
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