The purpose of this section is to define a boolean algebra and to determine some of the important properties of it.
A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complement ation, denoted by ', which satisfy the following axioms:
(1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws),
(2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) and a (b + c) = a b + a c, (the distributive laws),
(3) there exists 0 ∑B such that for each a ∑B, a + 0 = a, and there exists 1 ∑B such that for each a ∑ B, a 1 = a,
(4) for each a∑B, a + a' = 1 and a a' = 0.
If a + e = a for all a in B then 0 = 0 + e = e + 0 = e, so that there is exactly one element in B which satisfies the first half of axiom 3, namely 0. Similarly there is exactly one element in B which satisfies the second half of axiom 3, namely 1.
The O and 1 as defined above will be called the distinguished elements.
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-7842 |
| Date | 01 May 1963 |
| Creators | Eggert, Jr., Norman H. |
| Publisher | DigitalCommons@USU |
| Source Sets | Utah State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | All Graduate Theses and Dissertations |
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