Section 5  Booean Integers 0,I

Upon the small collection of zero (0) and unity (1) the binary operations of mod 2 addition and mod 2 multiplication have been defined. In this section a slight change, that will create a new addition operation in preparation of the the chapter on sets. The reader will find motivation for Boolean operations discussed below, in that chapter.

[5.1] (Binary Boolean Operations)
  (Addition)              0+0 = 0,    0+1 = 1,    1+0 = 1,    1+1 = 1
  (Multiplication)    0x0 = 0,    0x1 = 0,    1x0 = 0,    1x1 = 1.

Click here to see a discussion of the commutative, associative and distributive laws. The number 1 all by itself is a (trivial) commutative multiplicative group But The change prevents I from having an additive inverse.

The change makes the number 1 idempotent. A number is said to be idempotentunder a binary operation iff it produces back itself when combined with itself. Zero is always idempotent under both operations: 0+0 = 0 and 0x0 = 0. Unity is idempotent under multiplication: 1x1 = 1.