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Chapter 6 Order Relations and Structures
CSCI 115 Chapter 6 Order Relations and Structures
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§6.1 Partially Ordered Sets
CSCI 115 §6.1 Partially Ordered Sets
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§6.1 – Partially Ordered Sets
POSET A relation R on a set A is called a partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set or poset, and is denoted (A,R).
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§6.1 – Partially Ordered Sets
Dual Comparable Linear order (chain)
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§6.1 – Partially Ordered Sets
Theorem 6.1.1 If (A, 1) and (B, 2) are posets, then (A x B, ) is a poset where is defined by: (a, b) (a’, b’) iff a 1 a’ in A and b 2 b’ in B. (A x A, ) where 1 = 2 is called the product partial order
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§6.1 – Partially Ordered Sets
< a < b if a b and a b Lexicographic (dictionary) order Let (A, ) and (B, ) be posets. Then defined as (a, b) (a’, b’) iff a < a’ or a = a’ and b b’ is a partial order called the lexicographic or dictionary order.
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§6.1 – Partially Ordered Sets
Theorem 6.1.2 The digraph of a partial order has no cycle of length greater than 1
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§6.1 – Partially Ordered Sets
Hasse Diagram for (A, ) i) Draw digraph of ii) Delete all cycles of length 1 iii) Delete all edges implied by transitive property iv) Draw diagram with all edges pointing up and omit any arrows v) Replace circles with labeled points Hasse diagram gives a visual representation with all the implied components removed
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§6.1 – Partially Ordered Sets
Topological Sorting Linear order that is an extension of a partial order Typical notation: Many topological sortings may exist for a given partial order
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§6.1 – Partially Ordered Sets
Let (A, ) and (B, ) be posets. Let f:AB. f is called an isomorphism if: i) f is a 1-1 correspondence ii) a1, a2 A, a1 a2 iff f(a1) f(a2) In this case, we say (A, ) and (B, ) are isomorphic posets.
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§6.1 – Partially Ordered Sets
Theorem (Principle of correspondence) Let (A, ) and (B, ) be finite posets and f:AB be a 1-1 correspondence. Let H be the Hasse diagram of (A, ). Then: i) If f is an isomorphism and each label a of H is replaced by f(a), then H becomes a Hasse diagram for (B, ). ii) If H becomes a Hasse diagram for (B, ) when each label a of H is replaced by f(a), then f is an isomorphism.
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§6.2 Extremal Elements of Partially Ordered Sets
CSCI 115 §6.2 Extremal Elements of Partially Ordered Sets
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§6.2 Extremal elements of posets
Maximal Element aA is a maximal element of (A,R) if there does not exist cA s.t. a < c Minimal Element bA is a minimal element of (A,R) if there does not exist dA s.t. d < b Theorem 6.2.1 Let (A,) be a poset with A finite and non-empty. Then A has at least one maximal element, and at least one minimal element.
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§6.2 Extremal elements of posets
Procedure to find a topological sorting of a finite poset (A, ≤) Declare an array called SORT the size of |A| Choose a minimal element x of A Make x the next element in SORT Repeat steps 2 – 3 until A = {}
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§6.2 Extremal elements of posets
Greatest Element (Unit Element: 1) aA is a greatest element of (A,R) if xA x a. Least Element (Zero Element: 0) bA is a least element of (A,R) if xA b x. Theorem 6.2.2 A poset has at most one greatest element, and at most one least element.
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§6.2 Extremal elements of posets
Let (A, ) be a poset, with B A. Upper Bound (UB) aA is an upper bound of B if b a bB. Least Upper Bound (LUB) aA is a least upper bound of B if a is an upper bound for B, and a a’ whenever a’ is an upper bound of B. Lower Bound (LB) aA is a lower bound of B if a b bB. Greatest Lower Bound (GLB) aA is a greatest lower bound of B if a is a lower bound for B, and a’ a whenever a’ is a lower bound of B.
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§6.2 Extremal elements of posets
Theorem 6.2.3 Let (A, ) be a poset. Then a subset B of A has at most one LUB and at most one GLB.
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§6.2 Extremal elements of posets
Theorem 6.2.4 Suppose (A, ) and (B, ) are isomorphic posets under f:AB. Then: i) If a is a max (min) element of (A, ), then f(a) is a max (min) element of (B, ). ii) If a is a greatest (least) element of (A, ), then f(a) is a greatest (least) element of (B, ). iii) If a is an UB (LB, LUB, GLB) of (A, ), then f(a) is an UB (LB, LUB, GLB) of (B, ). iv) If every subset of (A, ) has a LUB (GLB), then every subset of (B, ) has a LUB (GLB).
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CSCI 115 §6.3 Lattices
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§6.3 – Lattices Lattice Poset (L, ) where every subset of 2 elements has a LUB and GLB Join of 2 elements a b = LUB ({a, b}) Meet of 2 elements a b = GLB ({a, b})
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§6.3 – Lattices Theorem 6.3.1 If (L1, 1) and (L2, 2) are lattices, then (L, ) is a lattice where L = L1 x L2 and is the product partial order Let (L, ) be a lattice. A non-empty subset S of L is called a sublattice of L if a b S and a b S a, b S
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§6.3 – Lattices Isomorphic Lattices
If f:L1 L2 is an isomorphism from the poset (L1, 1) to the poset (L2, 2), and if L1 and L2 are Lattices, then L1 and L2 are isomorphic lattices.
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§6.3 – Lattices Theorem 6.3.2 Theorem 6.3.3 – 6.3.7 in book
Let L be a lattice. a, b L we have: i) a b = b iff a b ii) a b = a iff a b iii) a b = a iff a b = b Theorem – in book We will not cover special types of lattices Bounded, distributive, complemented
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§6.4 Finite Boolean Algebras
CSCI 115 §6.4 Finite Boolean Algebras
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§6.4 – Finite Boolean Algebras
Theorem 6.4.1 If S1 = {x1, x2, …, xn} and S2 = {y1, y2, …, yn} are 2 finite sets with n elements, then the lattices (P(S1), ) and (P(S2), ) are isomorphic lattices. Consequently, the Hasse diagram of these lattices may be drawn identically.
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§6.4 – Finite Boolean Algebras
If the Hasse diagram of a lattice corresponding to a set with n elements is labeled by a sequence of 0s and 1s of length n, then the resulting lattice is called Bn.
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§6.4 – Finite Boolean Algebras
If x = a1a2…an and y = b1b2…bn are 2 elements of Bn, then the properties of Bn can be described by: i) x y iff ak bk for k = 1, 2, 3, …, n ii) x y = c1c2…cn where ck = min{ak, bk} iii) x y = d1d2…dn where dk = max{ak, bk}
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§6.4 – Finite Boolean Algebras
A finite lattice is called a Boolean Algebra if it is isomorphic to Bn for some nZ+ Theorem (modified) Dn is a boolean algebra iff n = p1p2…pk where the pi are all distinct primes Theorem and in book
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§6.5 Functions on Boolean Algebras
CSCI 115 §6.5 Functions on Boolean Algebras
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§6.5 – Fns on Boolean Algebras
Boolean Polynomials Let x1, x2, …, xn be a set of n variables. A Boolean Polynomial p(x1, x2, …, xn) in the variables xk is defined by the following: i) x1, x2, …, xn are all boolean polynomials ii) 0 and 1 are boolean polynomials iii) If p(x1, x2, …, xn) and q(x1, x2, …, xn) are both boolean polynomials in the variables xk, then p(x1, x2, …, xn) q(x1, x2, …, xn) and p(x1, x2, …, xn) q(x1, x2, …, xn) are also boolean polynomials iv) If p(x1, x2, …, xn) is a boolean polynomial, then so is If p(x1, x2, …, xn)’ v) Only polynomials generated by rules 1 – 4 are boolean polynomials
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§6.5 – Fns on Boolean Algebras
Manipulations Not responsible for manipulations Boolean Functions Similar to polynomial functions Accept arguments, and return values Evaluates to true or false
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§6.5 – Fns on Boolean Algebras
Schematic representations of boolean polynomials Used in circuitry, and other technical areas AND gates OR gates NOT inverters
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§6.5 – Fns on Boolean Algebras
The AND gate Accepts 2 arguments, and evaluates to true or false according to the logical rules for AND
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§6.5 – Fns on Boolean Algebras
The OR gate Accepts 2 arguments, and evaluates to true or false according to the logical rules for OR
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§6.5 – Fns on Boolean Algebras
The NOT inverter Accepts 1 argument, and evaluates to true or false according to the logical rules for NOT
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