Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Boolean Algebras Lecture 24 Section 5.3 Wed, Mar 22, 2006 Boolean Algebras In a Boolean algebra, we abstract the basic properties of sets and logic and make them the defining properties. A Boolean algebra has three operators + Addition (binary) Multiplication (binary) — Complement (unary) Properties of a Boolean Algebra Commutative Laws a+b=b+a ab=ba Associative Laws (a + b) + c = a + (b + c) (a b) c = a (b c) Properties of a Boolean Algebra Distributive Laws a + (b c) = (a + b) (a + c) a (b + c) = (a b) + (a c) Identity Laws: There exist elements, which we will label 0 and 1, that have the properties a+0=a a1=a Properties of a Boolean Algebra Complement Laws a +a = 1 a a = 0 Set-Theoretic Interpretation Let B be the power set of a universal set U. Interpret + to be , to be , and — to be complementation. Then what are the interpretations of 0 and 1? The identity and complement laws would be interpreted as A 0 = A, A 1 = A A Ac = 1, A Ac = 0 Logic Interpretation Let B be a collection of statements. Interpret + to be , to be , and — to be . Then what are the interpretations of 0 and 1? Again, look at the identity and complement laws. p 0 = p, p 1 = p p p = 1, p p = 0 Binary Interpretation Let B be the set of all binary strings of length n. Interpret + to be bitwise “or,” to be bitwise “and,” and — to be bitwise complement. Then what are the interpretations of 0 and 1? Look at the identity and complement laws. x | 0 = x, x & 1 = x x | x = 1, x & x = 0 Other Interpretations Let n be any positive integer that is the product of distinct primes. (E.g., n = 30.) Let B be the set of divisors of n. Interpret + to be gcd, to be lcm, and — to be division into n. For example, if n = 30, then a + b = gcd(a, b) a b = lcm(a, b) a = 30/a. Other Interpretations Then what are the interpretations of “0” and “1”? Look at the identity and complement laws. a + “0” = gcd(a, “0”) = a, a “1” = lcm(a, “1”) = a, a +a = gcd(a, 30/a) = “1”, a a = lcm(a, 30/a) = “0”. Connections How are all of these interpretations connected? Hint: The binary example is the most basic. Duality One can show that in each of the preceding examples, if we Reverse the interpretation of + and Reverse the interpretations of 0 and 1 the result will again be a Boolean algebra. Other Properties The other properties appearing in Theorem 1.1.1 on p. 14 can be derived as theorems. Double Negation Law a Idempotent Laws a+a=a aa=a Other Properties Universal Bounds Laws a+1=1 a0=0 DeMorgan’s Laws ( a b) a b ( a b) a b Other Properties Absorption Laws a + (a b) = a a (a + b) = a Complements of 0 and 1 0 = 1 1 = 0 Applications These laws are true for any interpretation of a Boolean algebra. For example, if a and b are integers, then gcd(a, lcm(a, b)) = a lcm(a, gcd(a, b)) = a If x and y are ints, then x | (x & y) == x x & (x | y) == x