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 |
| Rights | Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. |
Page generated in 0.0145 seconds