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Digital logic theory: How many distinct Boolean functions (i.e., different truth-table mappings) exist for four independent binary variables?

Difficulty: Easy

Correct Answer: 65536

Explanation:


Introduction / Context:
Counting the number of distinct Boolean expressions for a given number of variables is a foundational topic in digital logic and switching theory. Each distinct Boolean function corresponds to a unique mapping from all possible input combinations to a single binary output, regardless of how that function is written or simplified. This question checks your grasp of combinatorics applied to truth tables and Boolean functions.


Given Data / Assumptions:

  • Number of variables: 4 (say, A, B, C, D).
  • Each variable can take 2 values (0 or 1).
  • We count unique input-to-output mappings, not syntactic forms of expressions.


Concept / Approach:
The total number of input combinations for n binary variables is 2^n. A Boolean function assigns an output bit (0 or 1) to each input combination independently. Therefore, the number of distinct Boolean functions is 2 raised to the power of the number of input rows, i.e., 2^(2^n). This counts every possible truth table configuration exactly once.


Step-by-Step Solution:

Number of input rows for n = 4: 2^4 = 16.Each row can map to output 0 or 1 independently.Total distinct functions = 2^(number of rows) = 2^16.Compute: 2^16 = 65,536.


Verification / Alternative check:

For n = 1, there are 2^(2^1) = 4 functions (0, 1, identity, inversion). Extending to n = 2 yields 2^(2^2) = 16. The formula scales consistently, so for n = 4, 2^16 = 65,536 is correct.


Why Other Options Are Wrong:

16 and 256 underestimate by counting rows or small powers only.1024 (which is 2^10) is still far below 2^16.4096 (2^12) remains short; the correct count is 2^16, not 2^12.


Common Pitfalls:

Confusing 'number of minterms' (which is 16) with 'number of functions' (which is 2^16).Assuming each minterm must appear exactly once—functions may include, exclude, or combine minterms arbitrarily.


Final Answer:

65536
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