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Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Boolean Algebras Sajjad Department of Computer Science Swansea University December 08, 2008 Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Main Topic Boolean Rings. Boolean Algebras. Fields of Sets. Elementary Relations. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Rings Definition A ring is a set R together two binary operations +, and · defined on R such that: R with addition is an abelian group. Multiplication is an associative. Left and right distributive laws hold i.e., for all a; b; c belongs to R. a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · b) + (b · c) Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Boolean Ring Definition A Boolean ring R is a ring (with multiplicative identity) for which p = p2, for all p in R; that is, R consists only of idempotent elements. OR A Boolean ring R is a ring with unit in which every element is idempotent. Ring with unit means a ring with multiplicative identity 1 which is different from additive identity 0. and hence every Boolean ring contains 0 and 1. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Example The set 2X , of all functions from an arbitrary non-empty set X into 2 (here we denote the set of integers of modulo 2 by ordinary integer 2). Let f be a function which maps elements of X into 0 0 0 integer modulo 2, that is, Z2 = 0 ; 1 , where 0 represents all those elements whose reminder after dividing by 2 is 0. Similarly 10, represents all those elements whose reminder after dividing by 2 is 1. This means 0 and 1 in 2X , are functions defined point wise, for each x in X , by 0(x) = 00 and 1(x) = 10 Now we define the set 2X , as the set of all those functions f from X arbitrary set X into 2, i.e. 2 = f : X ! Z2 Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations If p and q (remember both are functions from X into Z2) belongs to 2X , then (p + q) and pq are defined by, (p + q) · (x) = p(x) + q(x) and (p · q)(x) = p(x) · q(x), for all x in X . The right sides of these equations refer to Z2 while left sides refer to 2X . Our task is to show that 2X is a Boolean ring i.e. 2X is a ring under point wise addition and multiplication. Proof. As 2X is a set of fucntion fron non empty set X to set of intergers of modulo 2, i.e. Z2, while is a ring with point wise addition and multiplication obiously. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations 2X has a multiplicative identity. Proof. Let 1 a function from 2X , then 0 0 1(x) = 1 , 1 belong to Z2. So for all p belong to 2X (p:1)(x) = p(x):(1) = p(x):10 = p(x) This implies 1 is a multiplicative identity in 2X . Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations All the elements of 2X are idempotent. Proof. Finally we need to show that all the elements of 2X are idempotent i.e p2 = p for all p in 2X . L:H:Sp2 = p· P2(x) = (p· )(x) = p(x)· (x) = [p(x)]2(1) 0 0 And as p(x) belongs to set of integers modulo 2, Z2 = 0 ; 1 So p(x) on the right hand side of (1) is either 00 or 10 the square of both is equal to themselves, i.e. if p(x) = 00 then 002 = 0 similarly if p(x) = 10 then 102 = 10. Hence (1) implies p2(x) = [p(x)]2 = p(x) hence all the elements of 2X are idempotent. Hence Proved that 2X is a Boolean ring. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Special Results of Boolean Rings Boolean rings has characteristics 2, that is (p + p = 0 for all p in A) Proof. Let p and q belongs to a Boolean ring A then consider (p + q)2 = (p + q) · (p + q) (p + q)2 = p2 + p · q + q · p + q2 As A is idempotent so (p + q)2 = (p + q) and p2 = p, q2 = q Put this in above equation we have: (p + q) = p + p · q + q · p + q p · q + q · p = 0 −−−−−−−−−−−−−−−−−−−−− (2) Now put p = q in Eq(2), we get p2 + p2 = 0 idempotence implies p +p = 0−−−−−−−−−−−−−−−−−−−−−−−−−−3) Proved Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Special Results of Boolean Rings Boolean ring is commutative. Proof. 2 implies p · q = −q · p and from (3) put p = −p on the left handside to get p · q = q · p −−−−−−−−−−−−−−− (4) Hence A is a Boolean ring. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Defining Boolean Algebra Definition A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join operators instead of the usual addition and multiplication operators. Or A Boolean algebra is a non-empty set A together with two distinct distinguished elements 0 and 1, two binary operations ^ and _ and a unary operation 0 satisfying the identities of Boolean rings in terms of meet, join, and complement; Those axioms are the following Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Defining Boolean Algebra 1 ,(00 = 1),(10 = 0) 2( p ^ 0 = 0), (p _ 1 = 1) 3( p ^ 1 = p), (p _ 0 = p) 4( p ^ p0 = 0), (p _ p0 = 1) 5( p00 = p) 6( p ^ p = p), (p _ p = p) 7( p ^ q)0 = p0 _ q0,(p _ q)0 = p0 ^ q0 8( p ^ q) = (q ^ p), (p _ q) = (q _ p) 9 p ^ (q ^ r) = (p ^ q) ^ r, p _ (q _ r) = (p _ q) _ r 10 p ^ (q _ r) = (p ^ q) _ (p ^ r), p _ (q ^ r) = (p _ q) ^ (p _ r) Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Important Result Every Boolean ring is a Boolean algebra, and voice versa. More precisely we can split it into two parts. (I) If R is a Boolea ring, for p and q in R, then we can define: 1 p ^ q = p · q 2 p _ q = p + q + p · q 3 p0 = 1 + p (II) If A is a Boolean algebra, for p and q in A, then we can define; 1 p · q = p ^ q 2 p + q = (p ^ q0) _ (p0 ^ q) Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Proof of some axioms of Boolean algebra by given definition of Boolean ring in (I) Proof. De Morgan's law, (p ^ q)0 = p0 _ q0 p0 _ q0 = p0 + q0 + p0q0 = p0 + q0(1 + p0), using def(3) and axiom(5), we get = p0 + 1 + q0p = 1 + p(1 + q0) again using def(3) and axiom(5), we get = 1 + pq = 1 + (p ^ q) = (p ^ q)0 Proved Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Proof of some axioms of Boolean algebra by given definition of Boolean ring in (I) Proof. Left distributive law,p ^ (q _ r) = (p ^ q) _ (p ^ r) (p ^ q) _ (p ^ r) = (p · q) _ (p · r) = (p · q) + (p · r) + (p · q · p · r) = (p · q) + (p · r) + (p · q · r), Idempotence p2 = p = p · (q + r + q · r) = p · (q _ r) = p ^ (q _ r) Proved Similarly we can prove all other axioms as well. Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Proof of some axioms of Boolean ring by given definition of Boolean algebra A in (II) Proof. The distingusihed elements 0 and 1 are obvious, here we have proved some other axioms Multiplication is associative, i.e.p · (q · r) = (p · q) · r (p · q) · r = p · (q ^ r) = p ^ (q ^ r) = (p ^ q) ^ r, Boolean algebra axiom = (p · q) · r proved Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Proof of some axioms of Boolean ring by given definition of Boolean algebra A in (II) Proof. The distingusihed elements 0 and 1 are obvious, here we have proved some other axioms Idempotence,i.e. for all p in R, p2 = p p · q = p ^ q, by replacing q by p we get p · q = p ^ p p2 = p Sajjad Department of Computer Science Swansea University Boolean Algebras Boolean Rings Boolean Algebra Field of Sets Elementry Relations Proof of some axioms of Boolean ring by given definition of Boolean algebra A in (II) Proof.
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