Boolean Algebra

The properties of Boolean algebra can be summarized in four basic rules.

1. Both binary operations have the property of commutativity, that is, order doesn't matter.
2. Each binary operation has an identity element associated with it. The universal set is the identity element for the operation of intersection, and the null set is the identity element for the operation of union. Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 4. Illustration by Hans & Cassidy. Courtesy of Gale Group.
3. each operation is distributive over the other.

This differs from the algebra of real numbers, for which multiplication is distributive over addition, a(b+c) = ab + ac, but addition is not distributive over multiplication, a+(bc) not equal (a+b)(a+c).

each element has associated with it a second element, such that the union or intersection of the two results in the identity element of the other operation.

This also differs from the algebra of real numbers. Each real number has two others associated with it, such that its sum with one of them is the identity element for addition, and its product with the other is the identity element for multiplication. That is, a + (-a) = 0, and a(1/a) = 1.