The Quine-McCluskey method is a systematic technique used to minimize Boolean functions, particularly for functions with several variables. It is a tabular method that simplifies Boolean expressions by eliminating redundant terms. Let's minimize the Boolean function $Y(a,b,c,d) = \sum m(0, 3, 4, 5, 6, 7, 11, 15)$ using the Quine-McCluskey method.

Step 1: Write Minterms in Binary Form

First, express each of the minterms in binary form (using 4 variables: $a, b, c, d$):

m(0) &= 0000 \\ m(3) &= 0011 \\ m(4) &= 0100 \\ m(5) &= 0101 \\ m(6) &= 0110 \\ m(7) &= 0111 \\ m(11) &= 1011 \\ m(15) &= 1111 \\ \end{aligned}$$ ### Step 2: Group Minterms by the Number of 1's Group the minterms based on the number of 1's in their binary representation: - **Group 0 (0 ones)**: $$m(0) = 0000$$ - **Group 1 (1 one)**: $$m(4) = 0100$$ - **Group 2 (2 ones)**: $$m(3) = 0011, m(5) = 0101, m(6) = 0110$$ - **Group 3 (3 ones)**: $$m(7) = 0111, m(11) = 1011$$ - **Group 4 (4 ones)**: $$m(15) = 1111$$ ### Step 3: Combine Minterms to Simplify Now, we combine minterms that differ by only one bit, replacing the differing bit with a dash (-). - Combine $$m(0) = 0000$$ and $$m(4) = 0100$$: $$0-00$$ - Combine $$m(4) = 0100$$ and $$m(6) = 0110$$: $$01-0$$ - Combine $$m(3) = 0011$$ and $$m(7) = 0111$$: $$0-11$$ - Combine $$m(5) = 0101$$ and $$m(7) = 0111$$: $$01-1$$ - Combine $$m(6) = 0110$$ and $$m(7) = 0111$$: $$011-$$ - Combine $$m(11) = 1011$$ and $$m(15) = 1111$$: $$1-11$$ ### Step 4: Combine Further if Possible Check if any of the terms from the previous step can be combined further: - Combine $$0-00$$ and $$0-11$$: not possible (differ by more than one bit) - Combine $$01-0$$ and $$01-1$$: $$01--$$ - Combine $$0-11$$, $$011-$$, and $$1-11$$: no further combination possible ### Step 5: Prime Implicants The remaining terms that could not be combined further are the prime implicants: - $$0-00$$ (covers $$m(0)$$ and $$m(4)$$) - $$01--$$ (covers $$m(4), m(5), m(6), m(7)$$) - $$011-$$ (covers $$m(6), m(7)$$) - $$0-11$$ (covers $$m(3), m(7)$$) - $$1-11$$ (covers $$m(11), m(15)$$) ### Step 6: Essential Prime Implicants Now, identify the essential prime implicants by covering all the original minterms: - $$01--$$ (covers $$m(4), m(5), m(6), m(7)$$) - $$0-00$$ (covers $$m(0)$$) - $$1-11$$ (covers $$m(11), m(15)$$) - $$0-11$$ (covers $$m(3)$$) These prime implicants cover all the minterms. ### Step 7: Final Simplified Expression Thus, the minimized Boolean expression is: $$Y(a, b, c, d) = \overline{a} \overline{b} + \overline{a}c + a \overline{c} \overline{d}$$ This is the minimized form of the given Boolean function using the Quine-McCluskey method. --- Would you like to see a breakdown of each step in more detail or have any questions? Here are 5 related questions for further exploration: 1. Can you explain how the Quine-McCluskey method handles don't-care conditions? 2. How does the Quine-McCluskey method differ from Karnaugh Maps for simplifying Boolean functions? 3. What are the computational limitations of the Quine-McCluskey method for large numbers of variables? 4. How are prime implicants used in other Boolean minimization techniques? 5. What is the significance of essential prime implicants in simplifying Boolean expressions? **Tip:** When simplifying Boolean functions with many variables, breaking the function down into smaller parts can make the process more manageable.