Boolean logic, or Boolean algebra as it is called today, was developed by an English mathematician, George Boole, in the 19th century. He based his concepts on the assumption that most quantities have two possible conditions - TRUE and FALSE. This is the same theory you were introduced to at the beginning of this chapter.
Throughout our discussions of fundamental logic gates, we have mentioned Boolean expressions. A Boolean expression is nothing more than a description of the input conditions necessary to get the desired output. These expressions are based on Boole's laws and theorems.
Boolean algebra is used primarily by design engineers. Using this system, they are able to arrange logic gates to accomplish desired tasks. Boolean algebra also enables the engineers to achieve the desired output by using the fewest number of logic gates. Since space, weight, and cost are important factors in the design of equipment, you would usually want to use as few parts as possible.
Figure 2-26 (view A), shows a rather complex series of gates. Through proper application of Boolean algebra, the circuit can be simplified to the single OR gate shown in view B. Figure 2-27 shows the simplification process and the Boolean laws and theorm used to accomplish it.
Figure 2-26. - Logic simplification: A. Complex series of gates; B. Simplified single OR gate
Figure 2-27. - Logic circuit simplification process.
LAWS AND THEOREMS
Each of the laws and theorems of Boolean algebra, along with a simple explanation, is listed below.
LAW OF IDENTITY - a term that is TRUE in one part of an expression will be TRUE in all parts of the expression (A = A or A = A).
COMMUTATIVE LAW - the order in which terms are written does not affect their value (AB = BA, A+B = B+A).
ASSOCIATIVE LAW - a simple equality statement A(BC) = ABC or A+(B+C) = A+B+C.
IDEMPOTENT LAW - a term ANDed with itself or ORed with itself is equal to that term (AA = A, A+A = A).
DOUBLE NEGATIVE LAW - a term that is inverted twice is equal to the term
COMPLEMENTARY LAW - a term ANDed with its complement equals 0, and a term ORed with its complement equals 1 (AA = 0, A+A = 1).
LAW OF INTERSECTION - a term ANDed with 1 equals that term and a term ANDed with 0 equals 0 (A· 1 = A, A · 0 = 0).
LAW OF UNION - a term ORed with 1 equals 1 and a term ORed with 0 equals that term (A+1 = 1, A+0 = A).
DeMORGAN'S THEOREM - this theorem consists of two parts: (1) AB = A + B and (2) A + B = A· B (Look at the fourth and eighth sets of gates in table 2-4).
DISTRIBUTIVE LAW - (1) a term (A) ANDed with an parenthetical expression (B+C) equals that term ANDed with each term within the parenthesis: A·(B+C) = AB+AC; (2) a term (A) ORed with a parenthetical expression ( B · C) equals that term ORed with each term within the parenthesis: A+(BC) = (A+B) · (A+C).
LAW OF ABSORPTION - this law is the result of the application of several other laws: A·(A+B) = A or A+(AB) = A.
LAW OF COMMON IDENTITIES - the two statements A·(A+B) = AB and A+AB = A+B are based on the complementary law.
Table 2-5 lists the Boolean laws and theorems and their equivalent statements.
Table 2-5. - Boolean Laws and Theorems
If you wish a more detailed study of Boolean algebra, we suggest you obtain Mathematics, Volume 3, NAVEDTRA 10073-A1.