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In the literature, different authors select different initial operations and axioms when defining an abstract Boolean algebra, but they are all easily seen to be equivalent to each other.To emphasise the abstract nature of these algebras, the symbols are often replaced with other symbols such as .] Clearly, every concrete Boolean algebra is an abstract Boolean algebra.Now let us turn from Boolean algebras to -algebras.
As with Boolean algebras, one can now define an to be a set with the indicated objects, operations, and relations, which obeys axioms 1-5.
Again, every concrete -algebra is an abstract one; but is it still true that every abstract -algebra is representable as a concrete one?
[Of course, some of these operations are redundant; for instance, intersection can be defined in terms of complement and union by de Morgan’s laws.
RA | Tags: boolean algebra, Loomis-Sikorski theorem, measure space, sigma-algebra, stone representation theorem, ultrafilters | by Terence Tao | 31 comments A (concrete) Boolean algebra is a pair , where X is a set, and is a collection of subsets of X which contain the empty set , and which is closed under unions , intersections , and complements . Because the is concretely represented as subsets of a space X, these relations automatically obey various axioms, in particular, for any , we have: to be an abstract set with the specified objects, operations, and relations that obey the axioms 1-4.
The largest cardinality among all the maximal independent sets is called the .
Of the many applications that arise, one in particular is in coding theory. We’ve discussed some basics of coding theory on this site as well.
Since then, I’ve been buried in graph theory texts, finding a wealth of fascinating topics to explore. If we added any other vertex to that set, it would be adjacent to some vertex already in there.
Of this article’s particular interest is finding all maximally independent sets in a graph using Boolean algebra. A few notes: (1) There are many maximal independent sets in a graph, and they may not all have the same cardinality. An independent set may be a maximal independent set without being the largest independent set in the graph.
It provides a set of rules (called Boolean logic) that are indispensable in digital computer-circuit and switching-circuit design.
Boolean operation is carried out with algebraic operators (called Boolean operators), the most basic of which are NOT, AND, and OR.