In 1854 George Boole introduced a systematic treatment of logic and developed for this purpose a systematic system now called boolean algebra. In 1938 C. E. Shannon introduced a two-valued Boolean algebra called switching algebra, in which he demonstrated that the properties of bistable electrical switching circuits can be represented by this algebra. in 1904 E. V. Huntington introduced a set of formal postulates defining boolean algebra.
Boolean algebra is an algebraic system defined on a set of elements B together with two binary operators + and ·.
Postulate 2 | x + 0 = x | x · 1 = x |
Postulate 5 | x + x' = 1 | x · x' = 0 |
Theorem 1 | x + x = x | x · x = x |
Theorem 2 | x + 1 = 1 | x · 0 = 0 |
Theorem 3, involution | (x')' = x | |
Postulate 3, commutative | x + y = y + x | x · y = y · x |
Theorem 4, associative | x + (y + z) = (x + y) + z | x · (y · z) = (x· y) · z |
Postulate 4, distributive | x · (y + z) = x · y + x · z | x+(y · z) = x + y · x + z |
Theorem 5, DeMorgan | (x + y)' = x'·y' | (x·y)' = x' + y' |
Theorem 6, absorption | x + (x · y) = x | x · (x + y) = x |
The duality principle states that a valid algebraic expression in boolean algebra remains valid if the operators and identity elements are interchanged. The rightmost two columns in the table above are duals.
Maintained by John Loomis, last updated 11 Sept 1999