**UNIT I – BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS**

**Boolean Variables & Truth Tables**

Boolean algebra differs in a major way from ordinary algebra in that Boolean constants and variables are allowed to have only two possible values, **0 or 1**.

Boolean 0 and 1 do not represent actual numbers but instead represent the state of a voltage variable, or what is called its logic level.

Some common representation of 0 and 1 is shown in the following diagram.

In Boolean algebra, there are three basic logic operations: AND ,OR, and NOT.

These logic gates are digital circuits constructed from diodes, transistors, and resistors connected in such a way that the circuit output is the result of a basic logic operation (OR, AND, NOT) performed on the inputs.

**Truth Table**

A truth table is a means for describing how a logic circuit's output depends on the logic levels present at the circuit's inputs.

In the following two-input logic circuit, the table lists all possible combinations of logic levels present at inputs A and B along with the corresponding output level X.

When either input A OR B is 1, the output X is 1. Therefore the "?" in the box is an OR gate.

**OR Operation**

The expression X = A + B reads as "X equals A **OR** B". The + sign stands for the OR operation, not for ordinary addition.

The OR operation produces a result of 1 when **any** of the input variable is 1.

The OR operation produces a result of 0 only when **all** the input variables are 0.

An example of three input OR gate and its truth table is as follows:

With the OR operation, 1 + 1 = 1, 1+ 1 + 1 = 1 and so on.

**AND Operation**

The expression X = A * B reads as "X equals A **AND** B".

The multiplication sign stands for the AND operation, same for ordinary multiplication of 1s and 0s.The AND operation produces a result of 1 occurs only for the single case when **all** of the input variables are 1.The output is 0 for any case where one or more inputs are 0

An example of three input AND gate and its truth table is as follows:

With the AND operation, 1*1 = 1, 1*1*1 = 1 and so on.

**NOT Operation**

The NOT operation is unlike the OR and AND operations in that it can be performed on a **single** input variable. For example, if the variable A is subjected to the NOT operation, the result x can be expressed as x = A' where the prime (') represents the NOT operation. This expression is read as: *x equals NOT A ** *

*x equals the inverse of A*

*x equals the complement of A*

Each of these is in common usage and all indicate that the logic value of x = A' is o pposite to the logic value of A. The truth table of the NOT operation is as follows:

*1'=0 because NOT 1 is 0** *

*0' = 1 because NOT 0 is 1*

The NOT operation is also referred to as inversion or complementation, and these terms are used interchangeably.

**NOR Operation**

__NOR__ and NAND gates are used extensively in digital circuitry. These gates combine the basic operations AND, OR and NOT, which make it relatively easy to describe then using Boolean algebra.

NOR gate symbol is the same as the OR gate symbol *except* that it has a small circle on the output. This small circle represents the inversion operation. Therefore the output expression of the two input NOR gate is:

X = (A + B)'

An example of three inputs OR gate **can** be constructed by a NOR gate plus a NOT gate:

**NAND Operation**

NAND gate symbol is the same as the AND gate symbol *except* that it has a small circle on the output. This small circle represents the inversion operation. Therefore the output expression of the two input NAND gate is:

X = (AB)'

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