Boole’s/Shannon’s Expansion Theorem provides a mechanism to decompose a Boolean function. The Theorem says that a Boolean expression ‘f’ of n Boolean variables (X1, X2, … Xn) can be decomposed as f(X1, X2, … Xn) = (X1 AND f(1, X2, …., Xn)) OR (X1′ AND f(0, X2, …., Xn)) Convince yourself that this decomposition is correct. (Nothing to submit for this) a. Can the OR in the above expression be replaced with an XOR? Justify your answer.
The Correct Answer and Explanation is:
Correct Answer:
No, the OR in the Shannon expansion theorem cannot be replaced with an XOR.
Explanation
Shannon’s Expansion Theorem is a foundational concept in Boolean algebra and digital logic design. It expresses a Boolean function f(X1, X2, ..., Xn) in terms of one of its variables and the function’s values when that variable is set to 1 and 0. The decomposition is:
f(X1, X2, …, Xn) = (X1 AND f(1, X2, …, Xn)) OR (X1′ AND f(0, X2, …, Xn))
This representation is known as the Shannon expansion or Boole’s expansion and is based on the principle of multiplexer logic, where one variable controls which sub-function gets selected. The OR operation plays a critical role in combining the two mutually exclusive conditions: one where X1 is 1 and one where X1 is 0.
The reason we cannot replace the OR with an XOR lies in the behavior of the XOR operation. XOR outputs 1 only when exactly one of the inputs is 1. On the other hand, the OR operation outputs 1 if at least one of the inputs is 1.
In Shannon’s expansion, the two terms — (X1 AND f(1, …)) and (X1′ AND f(0, …)) — are mutually exclusive, meaning that both cannot be true at the same time. When X1 is 1, X1′ is 0, and vice versa. Because of this, only one of the two AND terms can be non-zero at any time. Therefore, using XOR might appear to work in this specific scenario.
However, the definition and behavior of OR is more general and applies correctly regardless of function overlaps. If the two terms were not guaranteed to be mutually exclusive, XOR would produce incorrect results. To maintain consistency and correctness in all cases, the OR operator must be used.
In summary, even though XOR might seem to give the right result here, it is not logically equivalent to OR in this context, so the OR must not be replaced with XOR.
