Boolean_algebra - There are at least two distinct elements...

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Boolean Algebra Eng 151, Lecture 1 A Boolean Algebra B involves a set K and the operations Φ = {and, or, not} on K . Axioms (Postulates): basic rules that govern the manner in which algebraic operations can be used. Theorems: True statements that can be deduced (proven) from the axioms. Huntington’s Postulates: 1. a. Closure If a and b are in K, then a + b is in K b. If a and b are in K then ab is in K 2. a. Zero axiom There is an element 0 in K such that a + 0 = a b. Unit axiom There is an element 1 in K such that a 1 = a 3. a. Commutativity For all a and b in K , a + b = b + a b. For all a and b in K , ab = ba 4. a. Distributivity For all a , b , and c in K , a + bc = (a + b)(a + c) b. For all a , b , and c in K , a (b + c) = ab + ac 5. a-b. Inverse axioms For each a in K , there is an inverse or complement element a in K such that: 5a) a + a = 1 and 5b) a a =0 . 6.
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Unformatted text preview: There are at least two distinct elements in K . Useful Boolean Algebra Theorems: Theorem number Name Statement 7. a. Uniqueness of 0 and 1 The zero element 0 is unique b. The unit element 1 is unique 8. a. Idempotence For each a in K , a + a = a b. For each a in K , a a = a 9. a. Unit property For each a in K , a + 1 = 1 b. Zero property For each a in K , a 0 = 0 10. a. Absorption For all a and b in K , a + a b = a b. For all a and b in K , a (a + b) = a 11. Uniqueness of inverse For each a in K , the inverse a is unique 12. a. Associativity For all a , b , and c in K , (a + b) + c = a + (b + c) b. For all a , b , and c in K , (a b) c = a (b c) 13. a. De Morgan’s Laws For all a and b in K , ( ) b a b a ⋅ = + b. For all a and b in K , b a ab + = 14. Involution For each a in K , a a =...
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This note was uploaded on 01/19/2012 for the course ENG 151 taught by Professor Atkins during the Fall '10 term at University of Michigan.

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