De Morgan's Law Explained

The equality \((A + B + C)^I = A^I.B^I.C^I\) is a well-known principle in Boolean algebra and digital logic. This specific form demonstrates how to find the complement of a sum of variables, relating it to the product of their individual complements. This fundamental rule is precisely what De Morgan's Law describes.

De Morgan's Law Principle

De Morgan's Law is a set of two transformation rules in Boolean algebra that are essential for simplifying expressions and understanding logic gates. For any two variables, say A and B, the laws are stated as:

• Rule 1: The complement of a sum (OR operation) is equivalent to the product (AND operation) of the complements.

  \((A + B)^I = A^I.B^I\)

• Rule 2: The complement of a product (AND operation) is equivalent to the sum (OR operation) of the complements.

  \((A.B)^I = A^I + B^I\)

In the given question, the equality \((A + B + C)^I = A^I.B^I.C^I\) is an extension of the first rule of De Morgan's Law to three variables. The 'I' superscript denotes the complement of a variable or expression, similar to an apostrophe ('), an overbar (\(\overline{A}\)), or a prime symbol (\(A'\)). This law is crucial for converting AND operations to OR operations and vice-versa when complements are involved, which is vital in designing and analyzing logic circuits.

Analysis of Other Laws

To fully understand why De Morgan's Law is the correct answer, let's briefly examine the other options presented:

• Involution Law: This law states that if you complement a variable twice, you return to the original variable. It is represented as \((A^I)^I = A\). It signifies that a double negation cancels out, restoring the original state.
• Absorption Law: The absorption law helps simplify Boolean expressions by showing that a variable can "absorb" a term involving itself. The two forms are:
  • \(A + (A.B) = A\)
  • \(A.(A + B) = A\)
  This law reduces redundancy in expressions.
• Complementation Law: This law defines the relationship between a variable and its complement. It states that:
  • The OR of a variable and its complement is always True (or 1): \(A + A^I = 1\).
  • The AND of a variable and its complement is always False (or 0): \(A.A^I = 0\).
  These properties are fundamental to understanding how complements behave in Boolean algebra.

Conclusion on the Equality

Based on the definitions, the equality \((A + B + C)^I = A^I.B^I.C^I\) directly corresponds to the first rule of De Morgan's Law extended for three variables. This law is fundamental for transforming logical expressions and simplifying Boolean circuits, making it a cornerstone of digital electronics and formal logic.