The Structure of a Boolean Algebra Page: 2
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Postulate 1.7. There exist two unique elements in
namely 0 and I, such that if A is an element of then
0 < A and A < I.
Postulate 1.8. If A and B are elements of P, then
A < B if and only if B* O A < 0, and A < B if and only if
1 < A* U B.
Postulate 1.9. If A and B are elements of and if
A < B, then B < A«.
Postulate 1.1 is the reflexive property, Postulate 1.2
is the transitive property, and Postulate 1.3 is the anti-
symmetric property.
For the following theorems assume A, B, and C are
elements of 2-^. Unless a symbol is defined, its commonly
accepted meaning is assumed.
Theorem 1.1. A U A < A.
Proof: (AAJ B) < C if and only if A < C and B < C by
Postulate 1.6. Substituting A for B and A for C, (A U A) <A
if and only if A < A and A <" A. But A < A by Postulate 1.1.
Therefore, (A UA) <L A.
*
Theorem 1.2. A<A HA,
Proof; A < B A C if and only if A <1 B and A < C, by
Postulate 1.5. Substituting A for B and A for C, A < A C\ A
if and only if A < A and A < A, But A < A by Postulate 1.1.
Therefore, A < A H A,
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Bryant, June Anne. The Structure of a Boolean Algebra, thesis, August 1965; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130610/m1/6/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .