BINARY LOGIC - Be Engineer

BINARY LOGIC

Binary logic deals with variables that take on two discrete values and with operations that assume logical meaning. The two values the variables assume may be called by different names (true and false, yes and no, etc.), but for our purpose, it is convenient to think in terms of bits and assign the values 1 and 0. The binary logic introduced in this section is equivalent to an algebra called Boolean algebra. The formal presentation of Boolean algebra is covered in more detail in Chapter 2 . The purpose of this section is to introduce Boolean algebra in a heuristic manner and relate it to digital logic circuits and binary signals.

Definition of Binary Logic

Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc., with each variable having two and only two distinct possible values: 1 and 0. There are three basic logical operations: AND, OR, and NOT. Each operation produces a binary result, denoted by z.

AND: This operation is represented by a dot or by the absence of an operator. For example, x . y = z or xy = z is read “x AND y is equal to z.” The logical operation AND is interpreted to mean that z = 1 if and only if x = 1 and y = 1; otherwise z = 0. (Remember that x, y, and z are binary variables and can be equal either to 1 or 0, and nothing else.) The result of the operation x . y is z.

OR: This operation is represented by a plus sign. For example, x + y = z is read “x OR y is equal to z,” meaning that z = 1 if x = 1 or if y = 1 or if both x = 1 and y = 1. If both x = 0 and y = 0, then z = 0.

NOT: This operation is represented by a prime (sometimes by an overbar). For example, x_ = z (or x = z ) is read “not x is equal to z,” meaning that z is what x is not. In other words, if x = 1, then z = 0, but if x = 0, then z = 1. The NOT operation is also referred to as the complement operation, since it changes a 1 to 0 and a 0 to 1, i.e., the result of complementing 1 is 0, and vice versa.

Binary logic resembles binary arithmetic, and the operations AND and OR havesimilarities to multiplication and addition, respectively. In fact, the symbols used for .

Truth Tables of Logical Operations :

AND and OR are the same as those used for multiplication and addition. However, binary logic should not be confused with binary arithmetic. One should realize that an arithmetic variable designates a number that may consist of many digits. A logic variable is always either 1 or 0. For example, in binary arithmetic, we have 1 + 1 = 10 (read “one plus one is equal to 2”), whereas in binary logic, we have 1 + 1 = 1 (read “one OR one is equal to one”).

            For each combination of the values of x and y, there is a value of z specified by the definition of the logical operation. Definitions of logical operations may be listed in a compact form called truth tables. A truth table is a table of all possible combinations of the variables, showing the relation between the values that the variables may take and the result of the operation. The truth tables for the operations AND and OR with variables x and y are obtained by listing all possible values that the variables may have when combined in pairs. For each combination, the result of the operation is then listed in a separate row. The truth tables for AND, OR, and NOT are given in Table 1.8 . These tables clearly demonstrate the definition of the operations.