Table 6.2. Truth table showing the rule x ^. 1 = x.

6.2 Boolean Theorems

The interesting aspect about Boolean operators is that they usually follow simple rules, and these rules are very helpful in gaining insight and for solving problems. Consider the following mathematical example of multiplying two numbers 5 x 13 to get a result of 65. Using the rules of the multiplication operator the result could be obtained a little differently as 5x(10 + 3)=(5 x 10) + (5 x 3) = 65. Note that the second approach also gives the correct answer, but is based on the distributive rule of ax(b+c) =a x b +a x c. Like the distributive rule for multiplication, there are rules that work for the various Boolean operators. The rules for + and . are as follows.

Table 6.1. Boolean operator rules

The validity of these rules can be easily demonstrated using a truth table. For example consider the first rule for AND, which is x . 1 = x. Since x is a Boolean variable it can have two possible values [1 and 0] therefore it has to be shown that the rule statement works for both cases.

Table 6.2. Truth table showing the rule x ^. 1 = x.

By looking at the pattern in the first column and the pattern in the third column of the above table it is evident that the Boolean relationship x  ^. 1 = x is valid. Similarly all the other rules can be shown to be valid using truth tables.

Table 6.3. Truth table showing the rule x + (y ^. z)= (x + y)(x + z)

Once again it can be seen that the values in the fifth column are the same as the values in the last column. The equality of the values in both these columns therefore proves the validity of the rule x + (y ^. z) = (x + y)(x +z).