Boolean Logic

Computers are made up of digital circuits. Each component in a circuit has an input voltage that can be evaluated as True or False, or 1 or 0. Boolean logic is a way of describing the circuits inside a computer, and the same concept can be represented by a logic circuit diagram, a truth table, or a Boolean expression.

At a more advanced level, you will study more complex logic gates, although all circuits can ultimately be derived from the combination of a small subset of basic gates. Using a formal notation for describing the combination of logical operations will allow you to apply standard rules to simplify complex Boolean expressions. You will explore some standard circuits in detail, such as a full adder, which allows arithmetic addition, and a D-type flip-flop circuit, which is a very basic unit of memory.

What is Boolean Logic?

A Boolean variable is a variable that can only have one of two states. Often these states are referred to as True and False, but a Boolean variable can be used to represent any value pair, and it can be stored as a single bit.

Boolean algebra is a formal notation for describing logical relations. It was invented by George Boole (1815–1864), who demonstrated that all logical relations can be expressed as a combination of AND, OR, and NOT operations. In 1937, Claude Shannon applied Boole’s work to the design of switching circuits, and so Boolean logic became the foundation of all digital circuit design.

Logic Statements and Expressions

A logic statement is a statement that evaluates to either True or False. For example, “It is raining” is a logic statement: you can look out your window to determine whether the statement is True or False.

Boolean logic combines multiple logic statements that are either True or False into an expression that is either True or False. Consider the following logic statements:

It is sunny
It is hot

Now consider an expression that combines the two statements:

It is sunny AND it is hot

Because each of the two statements that make up the expression can be either True or False, there are four possible combinations to consider. We can show these combinations in a truth table, which will help us to evaluate the full expression.

To make a truth table for the individual expressions it is sunny and it is hot, you must first create a column for each statement and also create a row for each combination of True/False values: