Logic Of George Boole And Its Application To The Design Of Modern Computers

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rs. He died on December 8, 1864 in Ballintemple, Country Cork, Ireland.
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2. Boolean Algebra

Boolean Algebra is branch of mathematics having laws and properties similar to, but different from, those of ordinary high school algebra. Formally a Boolean algebra is a mathematical system consisting of a set of elements, which may be called B, together with two binary operations, which may be denoted by the symbols ⊕ and ⊗. These operations are defined on the set B and satisfy the following axioms:



1. ⊕ and ⊗ are both commutative operations. That is, for any elements x, y of the set B, it is true that x⊕Y = y⊕x and x⊗y = y⊗x.

2. Each of the operations ⊕ and ⊗ distributes over the other. That is, for any elements x, y, and z of the set B, it is true that x⊕ (y⊗z) = (x⊕y) ⊗ (x⊕z), and x⊗ (y⊕z) = (x⊗y) ⊕ (x⊗z).

3. There exists in the set B a distinct identity element for each of the operations ⊕ and ⊗. These elements are usually denoted by the symbols 0 and 1 such that 0 ≠ 1, and have the property that 0 ⊕x = x and 1 ⊗x = x for any element x in the set B.

4. For each element x in the set B there exists a distinct corresponding element called the complement of x, usually denoted by the symbol x’. With respect to the operations ⊕ and ⊗, the element x’ has the property that x⊕x’ = 1 and x ⊗x’ = 0.

A Boolean algebra may have other sets of axioms, all of which may be shown to be equivalent to those just given. The axioms given here are essentially those first published by the American mathematician Edward Huntington in Postulates for the Algebra of Logic (1904).

The English mathematician George Boole gave the first treatment of the subject in 1854. It is possible...