1.3.4 Boolean Function and its Conversion to Logic Circuit

A Boolean function is a mathematical expression built from binary variables and logical operators. It can be converted from an algebraic expression into a logic circuit.

Definition

A Boolean function is an expression formed with binary variables, the logical operators (OR, AND and NOT), parentheses and an equal sign.

Main Idea

A binary variable can take the value of 0 or 1. For a given value of the variables, the function can be either 0 or 1. A Boolean function can be transformed from an algebraic expression into a logic circuit composed of logic gates.

Core Concept

Structure of a Boolean Function A Boolean function is formed with: Binary variables Logical operators (OR, AND, and NOT) Parentheses An equal sign Values of a Boolean Function A binary variable can take the value of 0 or 1. For a given value of the variables, the function can be either 0 or 1.

Detailed Explanation

Example from the Book Consider the following Boolean function: F=x+yF=x+y The function F is equal to 0 if x=0x=0 and y=0y=0. For all the other combinations of xx and yy, the function will be equal to 1. This shows how a Boolean function depends on the values of its binary variables. The function produces a specific output (0 or 1) for each possible combination of input values. Conversion to Logic Circuit

A Boolean function can be transformed from an algebraic expression into a logic circuit composed of logic gates. This means that the operations in the Boolean expression (AND, OR, NOT) are directly implemented using the corresponding logic gates in a circuit.

Important Points

• A Boolean function is formed with binary variables, logical operators (OR, AND, NOT), parentheses, and an equal sign.
• A binary variable can take the value of 0 or 1.
• For a given value of the variables, the function can be either 0 or 1.
• A Boolean function can be transformed from an algebraic expression into a logic circuit composed of logic gates.
• Example: For F = x + y, the function equals 0 only when x = 0 and y = 0; for all other combinations, F = 1.

Remember

• Boolean function = binary variables + logical operators (OR, AND, NOT) + parentheses + equal sign.
• Binary variable 0 or 1.
• Function output 0 or 1 for a given set of variable values.
• Boolean function can be converted into a logic circuit.
• For F = x + y: F = 0 only when x = 0 and y = 0.

Common Mistakes

• Thinking a Boolean function can take values other than 0 and 1. Both the variables and the function take only 0 or 1.
• Forgetting that the + in a Boolean function stands for the OR operation, not normal addition.
• Missing parentheses when writing a Boolean function. Parentheses are part of the definition.
• Assuming a Boolean function is the same as a logic circuit. The function is the algebraic expression; the circuit is
• its implementation using logic gates.

Quick Revision

A Boolean function is an expression formed with binary variables, the logical operators (OR, AND and NOT), parentheses and an equal sign. A binary variable can take the value of 0 or 1. For a given value of the variables, the function can be either 0 or 1. As an example, consider the Boolean function F = x + y. The function F is equal to 0 if x = 0 and y = 0. For all the other combinations of x and y, the function will be equal to 1. A Boolean function can be transformed from an algebraic expression into a logic circuit composed of logic gates.

60-Second Revision

Boolean function = binary variables + AND, OR, NOT + parentheses + equal sign. Variables and function take only 0 or 1. F = x + y F = 0 only when x = 0 and y = 0; otherwise F = 1. Boolean function can be converted into a logic circuit. Warning: + means OR, not addition.