Textbook Problem

Let S = {O, 1}, and define operations + and · on S by the following tables:

a. Show that the elements of S satisfy the following properties:
(i) the commutative law for +
(ii) the commutative law for ·
(iii) the associative law for +
(iv) the associative law for ·
(v) the distributive law for + over ·
(vi) the distributive law for · over +
b. Show that 0 is an identity element for + and that 1 is an identity element for ·.
c. Define $\bar{0}=1$ and $\bar{1}=0$ . Show that for every a in S. $a+\bar{a}=1$ and $a\cdot \bar{a}=0$ . It follows from parts (a)—(c) that S is a Boolean algebra with the operations + and ·.