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Encyclopedia4U - Boolean algebra - Encyclopedia Article (1642 words) |
 | The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm). |
| Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, *) by defining a + b = (a ∧ ¬b) ∨ (b ∧ ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a * b = a ∧ b. |
| An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I. |
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