Boolean Algebra and Its Laws | Digital Electronics

Boolean algebra is considered as a system of mathematical logic which differs from both ordinary algebra and the binary number system. As for example, in Boolean 1 + 1 = 1 while in binary arithmetic the result is 10. Thus, Boolean algebra, though there are certain similarities, is a unique system.

In the Boolean system there are two constants, viz., 0 and 1. There are no fractional or negative numbers. In fact, every number is expressed either by 0 or by 1. Thus, if x = 1, then x ≠ 0 and if x = 0, then x ≠ 1. The system of Boolean algebra is applied to the solution of electronic circuit involving only two possible states.

The objectives of the use of Boolean algebra to any logic circuit are:

(i) To simplify the procedure required in solving logical problems, and

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(ii) To simplify any circuit to its fewest components necessary to perform the function.

AND, OR and NOT Operators:

1. AND Operator:

The AND function is defined in Boolean algebra by use of the dot and so it is similar to multiplication in ordinary algebra. As for example, A.B = C means that if A is true AND B is true then C will be true; otherwise C will be false. There are four different combinations of A and B as shown in Table 5.1.

It is to be noted that only when both A and B are 1, the output C will be 1. AND gates may have any number of inputs. Fig. 5.1 shows a four-input AND gate and its associated truth table.

And Laws:

Considering the AND symbol and its meaning the following three Boolean algebraic laws can be verified:

Let us consider equation (5.1) and apply it to a two-input AND gate as shown in Fig. 5.2. If A = 0 and the other input is 1 then the output will give zero [Fig. 5.2(a)]. But if A =1 and the other input is also 1 then the output is 1 [Fig. 5.2(b)]. Hence the output is always equal to the A input.

Let us now consider equation (5.2) and apply it to a two-input gate as shown in Fig. 5.3. It is found that if A becomes 0 or 1, the output will always be 0. Next we consider equation (5.3) and its realization in Fig. 5.3. It is seen that the output always takes the value of A.

2. OR Operator:

The OR operator is indicated by a plus (+) sign. Thus, A + B = C means that if A is true OR B is true then C will be true, otherwise (i.e., when both A and B are false) C will be false. There are four possible combinations as indicated in Table 5.2. It may be noted that 1 appears at the output in three of the four cases. The OR gate may have any number of inputs. A four-input OR gate along with its truth table is shown in Fig. 5.5.

The OR operator discussed above is called an inclusive OR, as it includes the case when both inputs are true. In fact, whenever an OR function is mentioned then inclusive OR is meant.

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The other type of OR function is known as the exclusive OR. Fig. 5.6 reveals the symbol and truth table for an exclusive OR gate. If either of the input but not both is true then the output is true, otherwise the output is false.

OR Laws:

There are several OR laws which can be verified by studying the OR gate.

These are given below:

The two possible cases of equation (5.4) are presented in Fig. 5.7.

If A is taken as either 0 or 1, the output is always 1. Equation (5.5) is established by means of Fig. 5.8. If A = 0, the output is 0 and if A = 1, the output is 1.

Thus, the output assumes the value of A. Equation (5.6) is next established by using Fig. 5.9. With A set to 0, the output is 0 and with A set to 1 the output is 1. Therefore, the output always equals A.

3. NOT Operator:

Operator is used to change the sense of an argument. The logic diagrams for the NOT operator are called inverters. With truth table the inverters are shown in Fig. 5.10. The circles in the figure represents the inversion and the triangle an amplifier.

NOT Laws:

The following laws of Boolean algebra become apparent when examining the inverter:

The above equations can be verified by considering Fig. 5.11. If the input is 0, then the output is 1 and if the input is 1, then the output is 0.

Another law that comes from the definition of an inverter is-

This law can be established by tor a considering Fig. 5.12. If A = 0 appears at the input of the first inverter then A̅ = 1/A̅ = and so the output of the second inverter A becomes 0. Thus, we can say that A inverted twice, i.e., A̅̿ is identical to A.

Laws of Boolean Algebra:

The Boolean algebra is a mathematical system. There are some fundamental laws of this algebra which are used to build a workable framework. Some of these laws have already been given in equations (5.1) to (5.11).

Others are given below:

(a) An important AND Law:

The law is expressed mathematically in the form-

The law can be verified by considering Fig. 5.13. If A = 0 then, A̅ = 0. That means the AND gate would have a 0 on one input and 1 on the other, resulting a 0 at the output. Similarly, if A = 1 then, A̅ = 0 and so the output of the AND gate becomes 0. Hence, we can say that A. A̅ = 0.