What is Boolean algebra?
Digital electronics are made up of special electronic circuits called integrated circuits that perform specific tasks, known as logical functions. These tasks can be written as simple mathematical equations, which we call logical expressions.
All digital circuits are built using basic building blocks called logic gates. We’ve talked about these gates in earlier articles. Each gate has a simple job, and we can describe that job using a mathematical equation.
Digital electronics work in a binary system, which means they only use two values: 0 or 1. For example, a 0 might mean "off," and a 1 might mean "on." Because of this, the inputs (what goes into the circuit) and outputs (what comes out of the circuit) are always either 0 or 1.
To work with these 0s and 1s in equations, we use a special type of math called Boolean algebra. It was created by a mathematician named George Boole. Boolean algebra gives us rules to simplify and solve these equations, making it easier to design and understand digital circuits.
Simplification of Algebraic Expressions
Diversity of Algebraic Expressions for the Same Logical Function
It’s important to note that a single logical function can be expressed by
multiple algebraic equations. In other words, there are many different Boolean
expressions that can represent the same logical function. As a result, it’s
possible to design electronic circuits with different structures that still
perform the same task. The main difference between these circuits lies in the
number of logic gates used.
For example, consider a logical function. It could be implemented using an
electronic circuit with 20 logic gates or another circuit with only 4 logic
gates, while both achieve the same functionality.
Reducing the Number of Logic Gates in Electronic Circuits
It makes sense to design electronic circuits using as few logic gates as possible. This helps lower manufacturing costs and makes the circuit smaller. But how do we achieve this? This is where Boolean algebra becomes essential. It provides powerful tools to simplify complex Boolean expressions into shorter, simpler ones while keeping the original logical function intact. To do this, Boolean algebra relies on a set of rules and principles, known as Boolean algebra laws, which we will explore in this article.
Logical Operations
Logical operations, also known as logical operators, are fundamental operations used in Boolean logic and digital electronics to process and manipulate Boolean values, which represent binary states (0 or 1). These operations allow us to combine logical propositions and obtain Boolean results. The most common logical operations are:
NOT Operation (Inverter):
The NOT operation, represented by a bar or line over a variable ( ), reverses the value of a logical proposition. It corresponds to the function of a NOT logic gate.
For example, if A is a logical proposition, then Ā represents its negation. If A is 1, Ā will be 0, and vice versa.
Logical OR Operation:
The OR operation, also known as disjunction, is a fundamental Boolean operation that combines two logical propositions to produce a new logical proposition. It is represented by the symbol (+).
Logical AND Operation:
The AND operation, also known as conjunction, is a fundamental operation in Boolean logic and digital electronics. The AND operation is typically represented by the symbol (.). It corresponds to the function of an AND logic gate.
Logical XOR Operation:
The XOR operation, also known as exclusive disjunction, exclusive OR, or binary sum, is a fundamental operation in Boolean logic and digital electronics. The XOR operation is typically represented by the symbol (⊕).
Theorems of Boolean Algebra:
Boolean algebra is based on a set of fundamental theorems that define the properties and relationships between logical operations (AND, OR, NOT, etc.). These theorems enable the simplification of complex Boolean expressions, optimization of logic circuits, and verification of the correctness of digital systems.
Studying Boolean logic teaches you that Boolean algebra relies on a set of basic rules. These rules are:
Postulates of Boolean Algebra
The postulates of Boolean algebra are the fundamental principles that form the foundation of this mathematical system. These basic rules define how logical operations (AND, OR, NOT, etc.) behave and are used to manipulate and simplify Boolean expressions in digital electronics.
Invariance Theorem
A + 1 = 1, A · 0 = 0: The invariance theorem in Boolean algebra states that when one input of a Boolean function is fixed at a constant value (0 or 1) and the other inputs take variable values, the output remains unchanged and retains the value of the fixed input.
Neutral Element Theorem
A · 1 = A, A + 0 = A: The neutral element theorem in Boolean algebra states that when one input
of a Boolean function is connected to the neutral constant of the
corresponding Boolean operation, the output value is always the same as the
value of the other input.
This theorem arises from the properties of the neutral elements of Boolean
operations (AND and OR). Specifically, the neutral element for the AND
operation is 1, and the neutral element for the OR operation is 0. If an input
is connected to the neutral element, it has no effect on the result of the
Boolean operation.
Idempotence Theorem
A · A = A, A + A = A: The idempotence theorem in Boolean algebra states that
when a Boolean function receives the same value for all its inputs, the output
value is always equal to that input value.
This theorem stems from the commutative and associative properties of Boolean
operations (AND and OR). Specifically, if all inputs have the same value, the
order and grouping of the operations do not affect the final result.
Complementarity Theorem
A · Ā = 0, A + Ā = 1: The complementarity theorem in Boolean algebra establishes a fundamental relationship between a proposition and its negation in the context of the Boolean operations AND and OR. It states that:
- For any proposition A, the disjunction (OR) of A and its negation Ā is always true (logical state 1).
- For any proposition A, the conjunction (AND) of A and its negation Ā is always false (logical state 0)..
Involution Theorem
The involution theorem in Boolean algebra, also known as the double negation theorem, states that applying negation to a Boolean proposition an even number of times results in the original proposition, while applying negation an odd number of times results in the negation of the original proposition:
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