We are tasked with simplifying the Boolean function $y(A, B, C) = \Sigma(0, 1, 3, 6, 7)$, which means that the function is expressed in terms of a sum of minterms for the given indices.

Step 1: Write the Minterms in Binary

First, represent the numbers (0, 1, 3, 6, 7) in binary based on the three variables $A$, $B$, and $C$.

0 & : (A'B'C') \\ 1 & : (A'B'C) \\ 3 & : (A'BC) \\ 6 & : (AB'C) \\ 7 & : (ABC) \end{aligned}$$ Thus, the function can be written as: $$y(A, B, C) = A'B'C' + A'B'C + A'BC + AB'C + ABC$$ ### Step 2: Simplify Using Boolean Algebra 1. **Group common terms**: We can group minterms where possible: $$y = A'B'C' + A'B'C + A'BC + AB'C + ABC$$ Notice that the first three terms have $$A'$$ in common: $$y = A'(B'C' + B'C + BC) + AB'C + ABC$$ 2. **Simplify the grouped terms**: We can simplify the expression inside the parentheses: $$B'C' + B'C + BC = B'(C' + C) + BC$$ Since $$C' + C = 1$$ (a basic Boolean identity), we get: $$B' + BC$$ Therefore, the simplified expression becomes: $$y = A'(B' + BC) + AB'C + ABC$$ 3. **Further simplifications**: Now distribute the terms: $$y = A'B' + A'BC + AB'C + ABC$$ Group the terms $$A'BC$$ and $$ABC$$: $$y = A'B' + AB'C + (A'BC + ABC)$$ Factor out $$C$$ from the last group: $$y = A'B' + AB'C + C(A'B + AB)$$ Simplify $$A'B + AB$$ (this is an XOR pattern, which simplifies to $$B$$): $$y = A'B' + AB'C + BC$$ ### Final Simplified Expression The simplified Boolean expression is: $$y(A, B, C) = A'B' + AB'C + BC$$ This is the minimal form of the function. --- Would you like further details or any clarifications? Here are 5 related questions for practice: 1. How would you represent the function if the given minterms were $$\Sigma(1, 2, 5, 6)$$? 2. What is the simplified Boolean expression for $$y(A,B,C) = \Sigma(2,3,6,7)$$? 3. How can you simplify the Boolean expression $$AB + A'B' + AB'$$ using Boolean algebra? 4. What are the key Boolean identities used to simplify logic expressions? 5. How do Karnaugh maps (K-maps) assist in simplifying Boolean functions? **Tip**: Always try to group common factors when simplifying Boolean expressions to reduce the number of terms.