The goal of this module is to enable learners to apply basic logic gates and Boolean expressions to digital circuits.

**Objectives**

A learner will be able to:

- Explain the difference between analog and digital quantities
- Give examples of binary numbers and describe their structure
- Give examples of hexadecimal and octal number systems and convert between binary, hexadecimal, and octal number systems
- Use a calculator to perform number system conversion and hexadecimal arithmetic
- Describe logic levels used in digital systems and analyze the characteristics of a pulse waveform
- Discuss the elements that make up a digital system

**Orienting Questions **

- What are the basic logic gates that are the basic building blocks for all logic circuits?
- What are the schematic symbols of the AND, OR, and NOT gates?
- What is Boolean Algebra?
- How do you apply Boolean Algebra to circuits containing AND, OR, and NOT gates?
- What are the schematic symbols for NAND and NOR gates?
- How do you apply Boolean Algebra to circuits containing NAND and NOR gates?

**Introduction**

The building blocks of all logic gates are the AND, OR, and NOT gates. In this module, we will explore the operation of each of these gates. We will also introduce Boolean Algebra, which describes the operation of all gates and gate combinations. At the end of the chapter, we will introduce the NAND(Not AND) and NOR(Not OR) gates and introduce their Boolean operators as well.

**AND, OR, NOT gates**

In this section we will learn about the operation of AND, OR, and NOT functions and their associated truth tables and Boolean expressions.

**AND gates**

All digital systems operate in binary, that is, they have two distinctive states or levels. These states or levels can be referred to as 0 or 1, high or low, on or off, true or false, the possibilities are endless. These states or levels describe a relationship between inputs and outputs. We can represent the most basic logic gates (AND and OR) as equivalent to the most basic types of circuits, that is, series and parallel.

**AND** gate operation is similar to a basic series circuit as shown in Figure 1 below. The inputs are represented by the switches sw1, and sw2 and the output is represented by the lamp. The “state” of sw1 and sw2 determines whether or not the lamp illuminates. It is easy to see that the only condition for which the lamp is on is that sw1 and sw2 must be closed. We can represent all possible states or conditions of sw1 and sw2 and the associated output through a **truth table**.

Figure 1 Series Circuit

Note that the number of input combinations depends on the number of switches in this case. We can write a general expression for this as follows

Number of to input states = 2^{n}

Where n = number of inputs

Therefore, a truth table for this setup would look like the following:

SW1 | SW2 | Bulb |

Open | Open | Off |

Open | Closed | Off |

Closed | Open | Off |

Closed | closed | on |

This series circuit configuration is the equivalent of the basic AND gate. The AND gate is then defined as a digital circuit whose output is “high” if and only if all inputs are “high”.

The schematic representation of an **AND** gate in digital circuits is:

A and B are the inputs to the gate, and X is the corresponding output.

The truth table for the 2 input AND gate in terms of Binary numbers is:

A | B | X |

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

0 = “off” or “low”

1 = “on” or “high”

**OR gates**

The next gate we will investigate is the operation of the OR gate. **OR** gate operation is similar to a basic parallel circuit as shown in Figure 2 below. The inputs are represented by the switches sw1, and sw2 and the output is represented by the lamp. The “state” of sw1 and sw2 determines whether or not the lamp illuminates. The conditions for which the lamp is on is that sw1 or sw2, or sw1 and sw2 must be closed. We can represent all possible states or conditions of sw1 and sw2 and the associated output through a truth table.

Figure 2: Basic parallel Circuit

Note that the number of input combinations depends on the number of switches in this case. We can write a general expression for this as follows:

Number of to input states = 2^{n}

Where n = number of inputs

Therefore, a truth table for this setup would look like the following:

SW1 | SW2 | Bulb |

Open | Open | Off |

Open | Closed | on |

Closed | Open | on |

Closed | closed | on |

This parallel circuit configuration is the equivalent of the basic OR gate. The **OR** gate is then defined as a digital circuit whose output is “high” if one or more inputs are “high”.

The schematic representation of an OR gate in digital circuits is:

Figure 3: OR Gate in Digital Circuits

A and B are the inputs to the gate, and X is the corresponding output.

The truth table for the 2 input OR gate in terms of Binary numbers is:

A | B | X |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

0 = “off” or “low”

1 = “on” or “high”

**NOT gates**

The last basic gate we will investigate is the NOT gate and its operation. The operation of the **NOT** gate is sometimes referred to as an **inverter**, that is, it inverts or changes a single logic level to the opposite value. So an input of “0” yields an output of “1”, and an input of “1” yields an output of “0”.

The schematic symbol for the inverter is as follows.

Figure 4: Schematic Symbol for the Inverter

The bubble on the end indicates an inversion of A!

An inverter only has one input and one output. That is, it operates only on one variable. The truth table for the inverter is:

A | X |

0 | 1 |

1 | 0 |

0 = “off” or “low”

1 = “on” or “high

The output of the inverter is the complement (opposite) of the input. It can be stated that output is the __not__ of the input. We will discuss the Boolean expression of the inverter in the next section.

**Boolean Algebra of AND OR And NOT gates**

Boolean algebra represents the mathematics of Digital Electronics Circuits. The operation of any logic gate or combination of gates can be described using Boolean algebra. In this section, we will look at Boolean algebra of the basic gates already discussed and address the basic Boolean rules (laws). We will also begin to draw circuits from un-simplified Boolean expressions.

**AND gate**

The **AND** gate represents Boolean multiplication, that is, the AND’ing or product of 2 inputs results in the output. Recall the truth table for the AND gate

A | B | X |

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

As already stated it has been shown that the AND operator is referred to as Boolean multiplication. The Boolean expression for this operation is given below.

**A*B = X**

**AB = X**

We will use the latter as the notation for Boolean multiplication from here on. The correct way to state this is “A and B = X”

When we take the product of two variables, the order in which the input variables are multiplied together does not affect the result. That is

**AB = BA Commutative Law for multiplication**

**OR gate**

The **OR** gate represents Boolean addition, that is, the OR’ing or sum of 2 inputs results in the output. Recall the truth table for the OR gate.

A | B | X |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

Remember we are not adding the two numbers, we are performing the “operation” between them. We are performing Boolean addition and not basic arithmetic.

As already stated it has been shown that the OR operator is referred to as Boolean addition (+). The Boolean expression for this operation is given below.

**A + B = X**

Properly stated “A or B = X”

When we take the Boolean sum of two variables, the order in which the input variables are summed does not affect the result. That is

**A + B = B + A Commutative Law for addition**

**NOT gate**

As already stated, the **NOT** gate represents Boolean complement, that is, the inversion of a single input results in the output. Recall the truth table for the inverter gate

A | X |

0 | 1 |

1 | 0 |

A = 1, X = 0

A = 0, X = 1

The Boolean expression for this operation is given

$\overline{A}$=X

Properly stated “A Not = X”

When we take the complement of a single variable, the output is the opposite of the input.

**Basic Boolean Rules**

Shown below in Figure 5 is how each gate would be represented in terms of its Boolean expression (as opposed to just X!). We will now write some Boolean rules associated with each of these basic gates. We will apply these rules throughout our study of combinatorial logic.

Figure 5: Basic Gates Representation

**Some basic Boolean Rules:**

- AB = BA
*Commutative Law for multiplication* - A + B = B + A
*Commutative Law for addition* - A ∙ 0 = 0
- A ∙ 1 = A
- A ∙ A = A
- A + 0 = A
- A + 1 = 1
- A + A = A
- A ∙ = 0
- A + = 0
- = A
- A + AB = A
- A + B = A + B
- (A+B)(A+C) = A + BC

**NAND and NOR Gates**

We can combine any of the basic gates AND, OR, and NOT to create new logic circuits with unique outputs. In this section we will investigate the operation of NAND (Not AND) and NOR (Not OR) gates and their associated schematics and Boolean expressions.

__NAND GATE__

__NAND GATE__

When we have an AND gate and follow it with an inverter, we have a **NAND** gate.

**Figure 6: NAND Gate**

By combining the AND and NOT gates this way, we have created a unique gate which has its own unique outputs for given inputs. The truth table would then look like this

A | B | AB | $\overline{AB}$ |

0 | 0 | 0 | 1 |

0 | 1 | 0 | 1 |

1 | 0 | 0 | 1 |

1 | 1 | 1 | 0 |

We can replace the above 2 gates with an equivalent NAND gate.

Figure 7: Schematic Symbol for NAND Gate

**Note:** the NAND schematic symbol is simply the AND gate with the negation symbol on its output

The truth table for the NAND gate is

A | B | X |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

__NOR GATE__

__NOR GATE__

When we have an OR gate and follow it with an inverter, we have a **NOR** gate.

** ** Figure 8: NOR Gate

By combining the OR and NOT gates this way, we have created a unique gate which has its own unique outputs for given inputs. The truth table would then look like this

A | B | A+B | $\overline{A+B}$ |

0 | 0 | 0 | 1 |

0 | 1 | 1 | 0 |

1 | 0 | 1 | 0 |

1 | 1 | 1 | 0 |

We can replace the above 2 gates with an equivalent NOR gate.

Figure 9: Schematic Symbol for NAND Gate

**Note:** the NOR schematic symbol is simply the OR gate with the negation symbol on its output.

The truth table for the NOR gate is

A | B | X |

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 0 |

**Boolean Expressions and Logic Circuits**

In order to analyze and troubleshoot digital circuits, it is necessary to be able to write Boolean expressions from logic circuits OR to draw circuits given only Boolean expressions.

Figure 10: 3 Input AND Gate Given Only 2 Input Gates.

This could be applied to any gates where n number of inputs is required and only 2 input gates are supplied. For example, if 4 inputs A, B, C, and D are needed, then just add an additional AND gate to circuit below.