Lattice and Boolean Algebra
Lattice and Boolean Algebra An algebraic system is defined by the tuple A,o1, …, ok; R1, …, Rm; c1, … ck, where, A is a non-empty set, oi is a function Api A, pi is a positive integer, Rj is a relation on A, and ci is an element of A. Lattice and Boolean Algebra
Lattice and Boolean Algebra The lattice is an algebraic system A, , , given a,b,c in A, the following axioms are satisfied: Idempotent laws: a a = a, a a = a; Commutative laws: a b = b a, a b = b a Associative laws: a (b c) = (a b) c, a (b c) = (a b) c Absorption laws: a (a b) = a, a (a b) = a Lattice and Boolean Algebra
Lattice and Boolean Algebra Lattice - Example Let A={1,2,3,6}. Let a b be the least common multiple Let a b be the greatest common divisor Then, the algebraic system A, , satisfies the axioms of the lattice. Lattice and Boolean Algebra
Lattice and Boolean Algebra Distributive Lattice The lattice A, , satisfying the following axiom is a distributive lattice 5. Distributive laws: a (b c) = (a b) (a c), a (b c) = (a b) (a c) Lattice and Boolean Algebra
Lattice and Boolean Algebra Examples distributive non-distributive Lattice and Boolean Algebra
Lattice and Boolean Algebra Complemented Lattice Let a lattice A, , have a maximum element 1 and a minimum element 0. For any element a in A, if there exists an element xa such that a xa = 1 and a xa = 0, then the lattice is a complemented lattice. Find complements in the previous example Lattice and Boolean Algebra
Lattice and Boolean Algebra Let B be a set with at least two elements 0 and 1. Let two binary operations and , and a unary operation are defined on B. The algebraic system B, , , , 0,1 is a Boolean algebra, if the following postulates are satisfied: Idempotent laws: a a = a, a a = a; Commutative laws: a b = b a, a b = b a Associative laws: a (b c) = (a b) c, a (b c) = (a b) c Absorption laws: a (a b) = a, a (a b) = a Distributive laws: a (b c) = (a b) (a c), a (b c) = (a b) (a c) Lattice and Boolean Algebra
Lattice and Boolean Algebra Involution: Complements: a a = 1, a a = 0; Identities: a 0 = a, a 1 = a; a 1 = 1, a 0 = 0; De Morgan’s laws: Lattice and Boolean Algebra
Huntington’s Postulates To verify whether a given algebra is a Boolean algebra we only need to check 4 postulates: Identities Commutative laws Distributive laws Complements Lattice and Boolean Algebra
Lattice and Boolean Algebra Example prove the idempotent laws given Huntington’s postulates: a = a 0 = a aa = (a a) (a a) = (a a) 1 = a a Lattice and Boolean Algebra
Models of Boolean Algebra Boolean Algebra over {0,1} B={0,1}. B, , , , 0,1 Boolean Algebra over Boolean Vectors Bn = {(a1, a2, … , an) | ai {0,1}} Let a=(a1, a2, … , an) and b = (b1, b2, … , bn) Bn define a b = (a1 b1, a2 b2, … , an bn) a b = (a1 b1, a2 b2, … , an bn) a=(a1, a2, … , an) then Bn, , , , 0,1 is a Boolean algebra, where, 0 = (0,0, …, 0) and 1 = (1,1, …, 1) Boolean Algebra over Power Set Lattice and Boolean Algebra
Lattice and Boolean Algebra Examples B3 P({a,b,c}) {n | n|30} Lattice and Boolean Algebra
Isomorphic Boolean Algebra Two Boolean algebras A, , , , 0A,1A and B, , , , 0B,1B are isomorphic iff there is a mapping f:AB, such that for arbitrary a,b A, f(ab) = f(a)f(b), f(a b) = f(a) f(b), and f(a) = f(a) f(0A ) = 0B and f(1A ) = 1B An arbitrary finite Boolean algebra is isomorphic to the Boolean algebra Bn, , , , 0,1 Question: define the mappings for the previous slide. Lattice and Boolean Algebra
Lattice and Boolean Algebra De Morgan’s Theorem De Morgan’s Laws hold These equations can be generalized Lattice and Boolean Algebra
Lattice and Boolean Algebra Definition Let Bn, , , , 0,1 be a Boolean algebra. The variable that takes arbitrary values in the set B is a Boolean variable. The expression that is obtained from the Boolean variables and constants by combining with the operators , , and parenthesis is a Boolean expression. If a mapping f:Bn B is represented by a Boolean expression, then f is a Boolean function. However, not all mappings f:Bn B are Boolean functions. Lattice and Boolean Algebra
Lattice and Boolean Algebra Theorem Let F(x1, x2, …, xn) be a Boolean expression. Then the complement of the complement of the Boolean expression F(x1, x2, …, xn) is obtained from F as follows Add parenthesis according to the order of operations Interchange with Interchange xi with xi Interchange 0 with 1 Example Lattice and Boolean Algebra
Lattice and Boolean Algebra Principle of Duality In the axioms of Boolean algebra, in an equation that contains , , 0, or 1, if we interchange with , and/or 0 with 1, then the other equation holds. Lattice and Boolean Algebra
Dual Boolean Expressions Let A be a Boolean expression. The dual AD is defined recursively as follows: 0D = 1 1D = 0 if xi is a variable, then xiD = xi if A, B, and C are Boolean expressions, and A = B C, then AD = BD CD if A, B, and C are Boolean expressions, and A = B C, then AD = BD CD if A and B are Boolean expressions, and A = B, then Lattice and Boolean Algebra
Lattice and Boolean Algebra Examples Given xy yz = xy yz xz the dual (x y)(y z) = (x y)(y z)(x z) Consider the Boolean algebra B={0,1,a,a} check if f is a Boolean function. f(x) = xf(0) xf(1) f(x) = x a x 1 f(a) = a a a 1 = a x f(x) a 1 a Lattice and Boolean Algebra
Lattice and Boolean Algebra Logic Functions Let B = {0,1}. A mapping Bn B is always represented by a Boolean expression–a two-valued logic function. f g = h f(x1,x2,…,xn) g(x1,x2,…,xn) = h(x1,x2,…,xn) f = g f(x1,x2,…,xn) = g(x1,x2,…,xn) x y f g fg fg f g 1 Example Lattice and Boolean Algebra
Lattice and Boolean Algebra Logical Expressions Constants 0 and 1 are logical expressions Variables x1,x2,…,xn are logical expressions If E is a logical expression, then E is one If E1 and E2 are logical expressions, then (E1 E2) and (E1 E2) are also logical expressions The logical expressions are obtained by finite application of 1 - 4 Lattice and Boolean Algebra
Evaluation of logical Expressions An assignment mapping :{xi} {0,1} (i = 1, … , n) The valuation mapping |F| of a logical expression is obtained: |0| = 0 and |1| = 1 If xi is a variable, then | xi | = (xi) If F is a logical expression, then |F| = 1 |F| = 0 If F and G are logical expressions, then |F G| = 1 (|F| = 1 or |G| = 1) If F and G are logical expressions, then |F G| = 1 (|F| = 1 and |G| = 1) Example: F:x y z (x) = 0, (y) = 0, (z) = 1 Lattice and Boolean Algebra
Equivalence of Logic Expressions Let F and G be logical expressions. If |F| = |G| hold for every assignment , then F and G are equivalent ==> F G Logical expressions can be classified into 22n equivalence classes by the equivalence relation () Lattice and Boolean Algebra