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OU k 105%, 368 (384 of 490) Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and4, respectively). These laws are used to define an abstract mathematical structure called a Boolean algebra, which is named after the mathematician George Boole (1815-1864) 15.2 BASIC DEFINITIONS Let B be a nonempty set with two bi 0 and 1. Then B is called a Boolean algebra if the following avioms hold where a, b, c are any elements in B nary operatns + and *, a unary operation , and two distinct elements IB1 Commutative laws (la) a +b (2a) (3a) b+a [B21 Distributive laws IB3l Identity laws: IB41 Complement laws: (lb) ab ba (2b) (3b) a 1 a (4b) aa'0 a + (b * c) = (a + b) * (a+c) a * (b + c) = (a * b) + (a*c) a+0=a (4a) a + a'=1 We will ometimes designate a Boolean algebra by (B + ,0, 1) when we want to emphasize its six parts. We say 0 is the zero element, I, is the unit element, and a' is the complement of a. We will usually drop the symbol * and use juxtaposition instead. Then (2b) is written ab + c)ab +ac which is the familiar algebraic identity of rings and fields. However, (2a) becomes a + be la + ba c). which is certainly not a usual identity in algebra The operations +. , and are called sum, product, and complement, respectively. We adopt the usual convention that, unless we are guided by parentheses, has precedence over *, and * has precedence over + For example, a + b sc means a +(b+ c) and not (a + b) c a b' means a * (b) and not (a *b) Of course when a +bsc is written a + be then the meaning is clear. EXAMPLE 15.1 (a) Let B (0, 1), the set of bits (binary digits), with the binary operations of + and e and the unary operation defined by Fig. 15-1. Then B is a Boolean algebra. (Note simply changes the bit, i.e., I0 and -1) 368