1. Boolean Algebra and Computer Logic Mathematical Structures for Computer Science Chapter 7.1 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesBoolean Algebra Structure

2. Section 7.1Boolean Algebra Structure1 Models or Abstractions We frequently use models or abstractions to capture properties that may be common to different instances or applications of the models. For example, graphs and trees are abstractions of many real-world problems. When we reason about a graph structure we are able to prove theorems and develop techniques that can be applied to a variety of problems. Mathematical structures—abstract sets of objects, together with operations on or relationships among those objects that obey certain rules—are often used as models that express the similarities between different instances.

3. Section 7.1Boolean Algebra Structure2 Definition and Properties A Boolean algebra is a set B on which are defined two binary operations and and one unary operation and in which there are two distinct elements 0 and 1 such that the following properties hold for all x, y, z  B: x + y = y + x (commutative properties) (x + y) + z = x + (y + z) (associative properties) x + (y * z) = (x + y) * (x + z) (distributive properties) x + 0 = x (identity properties) x + x = 1 (complement properties) And x * y = y * x (commutative properties) (x* y) * z = x * (y * z) (associative properties) x * (y + z) = (x * y) + (x * z) (distributive properties) x * 1 = x (identity properties) x * x = 0 (complement properties)

4. Section 7.1Boolean Algebra Structure3 Definition and Properties The formalization of a Boolean algebra structure helps us focus on the essential features common to all examples of Boolean algebras, and we can use these features—these facts from the definition of a Boolean algebra—to prove other facts about Boolean algebras. We denote a Boolean algebra by [B, +, ,, 0, 1]. Using the basic definition of Boolean algebra it is possible to prove other properties. Boolean algebra: forms the basis for electronics today.

5. Section 7.1Boolean Algebra Structure4 Definition and Properties For example, The idempotent property x + x = x holds in any Boolean algebra because: x + x = (x + x)  1// identity property = (x + x)  (x + x)// definition of x + x = x + (x  x)// distributive property = x + 0// complement property for  = x // identity property Ordinary arithmetic is not a Boolean algebra because the property x + x = x does not hold for ordinary numbers and ordinary addition unless x is zero. The idempotent property also holds for  : x  x

6. Section 7.1Boolean Algebra Structure5 Definition and Properties Hints for proving Boolean algebra equalities: Usually the best approach is to start with the more complicated expression and try to show that it reduces to the simpler expression. Think of adding some form of 0 (like x  x ) or multiplying by some form of 1 (like x + x ). Remember the distributive property of addition over multiplication—easy to forget because it doesn’t look like arithmetic. Remember the idempotent properties x + x = x and x  x = x.

7. Section 7.1Boolean Algebra Structure6 Definition and Properties For x an element of a Boolean algebra B, the element x is called the complement of x. The complement of x satisfies: x + x = l and x  x = 0 THEOREM ON THE UNIQUENESS OF COMPLEMENTS For any x in a Boolean algebra, if an element x 1 exists such that x + x 1 = 1 and x  x 1 = 0, then x 1 = x.

8. Section 7.1Boolean Algebra Structure7 Isomorphic Boolean Algebras Two instances of a structure are isomorphic if there is a bijection (called an isomorphism) that maps the elements of one instance onto the elements of the other so that important properties are preserved. If two instances of a structure are isomorphic, each is a mirror image of the other, with the elements simply relabeled. The two instances are essentially the same. Therefore, we can use the idea of isomorphism to classify instances of a structure, lumping together those that are isomorphic.

9. Section 7.1Boolean Algebra Structure8 Isomorphic Boolean Algebras Suppose we have two Boolean algebras, [B, +, ,, 0, 1] and [b, &, *, , φ, 1]. If x is in B, x is the result of performing on x the unary operation defined in B. If z is an element of b, z  is the result of performing on z the unary operation defined in b. To prove isomorphism, we need a bijection f from B onto b. Then f must preserve in b the effects of the various operations in B. There are three operations, so we use three equations to express these preservations.

10. Section 7.1Boolean Algebra Structure9 Isomorphic Boolean Algebras DEFINITION: ISOMORPHISM FOR BOOLEAN ALGEBRAS Let [B, +, ,, 0, 1] and [b, &, *, ′, ϕ, 1 ] be Boolean algebras. A function B  b is an isomorphism from [B, +, ,, 0, 1] to [b, &, *, , ϕ, 1 ] if: 1. f is a bijection 2. f (x + y) = f (x) & f (y) 3. f (x. y) = f (x) * f (y) 4. f (x′) = ( f (x)) 

11. Section 7.1Boolean Algebra Structure10 Isomorphic Boolean Algebras THEOREM ON FINITE BOOLEAN ALGEBRAS Let B be any Boolean algebra with n elements. Then n = 2m for some m, and B is isomorphic to  ({1, 2,..., m}). The theorem above gives us two pieces of information: The number of elements in a finite Boolean algebra must be a power of 2. Finite Boolean algebras that are power sets are—in our lumping together of isomorphic things—really the only kinds of finite Boolean algebras.