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Presentation transcript:

Background material

Relations A relation over a set S is a set R µ S £ S A relation R is: We write a R b for (a,b) 2 R A relation R is: reflexive iff 8 a 2 S . a R a transitive iff 8 a 2 S, b 2 S, c 2 S . a R b Æ b R c ) a R c symmetric iff 8 a, b 2 S . a R b ) b R a anti-symmetric iff 8 a, b, 2 S . a R b ) :(b R a)

Relations A relation over a set S is a set R µ S £ S A relation R is: We write a R b for (a,b) 2 R A relation R is: reflexive iff 8 a 2 S . a R a transitive iff 8 a 2 S, b 2 S, c 2 S . a R b Æ b R c ) a R c symmetric iff 8 a, b 2 S . a R b ) b R a anti-symmetric iff 8 a, b, 2 S . a R b ) :(b R a) 8 a, b, 2 S . a R b Æ b R a ) a = b

Partial orders An equivalence class is a relation that is: A partial order is a relation that is:

Partial orders An equivalence class is a relation that is: reflexive, transitive, symmetric A partial order is a relation that is: reflexive, transitive, anti-symmetric A partially ordered set (a poset) is a pair (S,·) of a set S and a partial order · over the set Examples of posets: (2S, µ), (Z, ·), (Z, divides)

Lub and glb Given a poset (S, ·), and two elements a 2 S and b 2 S, then the: least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S . (a · d Æ b · d) ) c · d greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S . (d · a Æ d · b) ) d · c

Lub and glb Given a poset (S, ·), and two elements a 2 S and b 2 S, then the: least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S . (a · d Æ b · d) ) c · d greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S . (d · a Æ d · b) ) d · c lub and glb don’t always exists:

Lub and glb Given a poset (S, ·), and two elements a 2 S and b 2 S, then the: least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S . (a · d Æ b · d) ) c · d greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S . (d · a Æ d · b) ) d · c lub and glb don’t always exists:

Lattices A lattice is a tuple (S, v, ?, >, t, u) such that: (S, v) is a poset 8 a 2 S . ? v a 8 a 2 S . a v > Every two elements from S have a lub and a glb t is the least upper bound operator, called a join u is the greatest lower bound operator, called a meet

Examples of lattices Powerset lattice

Examples of lattices Powerset lattice

Examples of lattices Booleans expressions

Examples of lattices Booleans expressions

Examples of lattices Booleans expressions

Examples of lattices Booleans expressions