# Boolean Algebra

Boolean algebra (named after George Boole, who first formulated a "logical algebra") is the mathematics of digital systems. A variable is a symbol used to represent a logical quantity.

The complement is the inverse of a variable and is indicated by a bar over the variable. The double complement of a variable is always equal to the variable.

The basic rules of Boolean algebra and their relation to the basic gates are illustrated in the diagram.

# Deriving Boolean Expression

Boolean algebra provides a way to express the operation of a logic circuit formed by logic gates so that the output can be determined by various combinations of input values. For n input variables, there are 2n possible binary input combinations. For each input binary combination there is only one possible binary value for the output.

For every logic circuit there is a Boolean expression. To derive the Boolean expression for a given logic circuit, begin at the left-most input, writing the expression for each gate and working towards the final output. Using this rule, derive a Boolean expression for the circuit in the diagram.

# Standard Form of Boolean Expressions

When two or more product terms are summed by Boolean addition, the resulting expression is a sum-of-product (SOP). When two or more sum terms are multiplied, the resulting expression is a product-of-sum (POS). All Boolean expressions, regardless of their form, can be converted into either of these two standard forms.

The diagram shows gate equivalents of standard SOP and POS expressions.

# De Morgan's Theorems

De Morgan's theorems are extremely useful in simplifying Boolean expressions in which a product or sum of variables is inverted. These theorems provide verification of the equivalence of the NAND and negative-OR gates as well as of the NOR and negative-AND gates.

The diagram illustrates De Morgan's theorems. Note the equality of the two output columns in the truth table. This shows that equivalent gates perform the same logic function.

# Equivalent Gates

The diagram shows the standard and the alternative equivalent symbol for each logic gate and the corresponding De Morgan's theorem. The alternative symbols are obtained as follows:

1. Each input and output of the standard symbol is inverted by adding bubbles to input and output lines that do not have bubbles or by removing any bubbles that are already there.

2. The operation symbol is changed from AND to OR or from OR to AND.

# Combinational Logic

Generally, a circuit is considered a combinational logic circuit if its output is either LOW or HIGH in response to a specific combination of input signals. The sequence in which the inputs are applied is not important.

The circuit in the diagram monitors the liquid level in each tank and indicates when the level in any two of the tanks drops below one third. If the liquid in two of the tanks drops below a critical level, regardless of the sequence in which this happens, the output always goes HIGH and the LED indicates a problem.

# Logic Implementation with PALs

PALs are used to produce specific custom combinational logic functions.

Play and learn to implement combinational logic with a PAL. Remember that the fusible links are left intact to connect the desired variables or their complements to the appropriate AND gate inputs. Fusible links are opened where a variable or its complement is not used in a given product term. In the diagram the intact fusible links are indicated by a dark green X.

# Combinatorial Logic Functions – Code Converter

There are a number of fundamental combinational circuits, referred to as functional blocks, that are very useful in designing larger digital circuits. These circuits include code converters, encoders, decoders, multiplexers as well as adders-subtractors, and multipliers which use binary and BCD codes.

Binary coded decimal (BCD) is a way to express each of the decimal digits as a binary code of four digits. There are only ten code groups in the BCD system, so it is very easy to convert between decimal and BCD. This code is also called 8421 code. The designation 8421 indicates the binary weights of the four bits (23, 22, 21, 20).

# BCD-to-7-Segment Decoder

Digital readouts in electronic equipment often use light-emitting diodes (LEDs). Each digit of the readout is formed from seven segments each consisting of one LED that can be illuminated by digital signals.

A BCD-to-7-segment decoder accepts BCD code at its inputs and provides outputs to drive a 7-segment display and produce a decimal readout. The seven outputs of the decoder (a, b, c, d, e, f, g) select the appropriate segments in the display, as shown in the diagram. This LED arrangement with a common cathode requires the driver to output a HIGH voltage to activate a segment.

# Example Application

Arrange the appropriate basic functional blocks to form a block diagram of a simple calculator. Remember that in digital systems the information that is being processed is usually presented in binary form but input data and result are normally in decimal form.

Play and learn with the calculator to observe data conversion in the case of addition/subtraction.

# Time Division Multiplexing

The diagram illustrates an application where digital data from three computers are sent out along a single line to three peripheral devices at another location. During the first time interval, data bits from computer A1 are connected to the output line and transmitted to the demultiplexer that connects them to scanner. During the second time interval, computer A2 sends data to an inkjet printer. During the third time interval, computer A3 sends data to a laser printer. After this the sequence repeats.

Because the time is divided up among several pieces of equipment where each has its turn to send and receive data, this process is called time division multiplexing.