1. **Problem Statement:** We are given several Boolean functions and minterm expressions: - $D = A \oplus B \oplus C$ - $E = A'BC + AB'C$ - $F = ABC' + (A' + B')C$ - $G = ABC$ - $F_1 = \Sigma(1,4,6)$ - $F_2 = \Sigma(3,5)$ - $F_3 = \Sigma(2,4,6,7)$ - $F_1(A,B,C,D) = \Sigma(1,2,5,7,8,10,11,13,15)$ We will analyze and simplify these Boolean functions step-by-step. 2. **Recall Boolean Algebra Rules:** - $A' = \text{NOT } A$ - $+$ means OR - Concatenation means AND - $\oplus$ means XOR (exclusive OR) - Minterms $\Sigma(m_1,m_2,...)$ represent sum of minterms where the function is 1. 3. **Simplify $D = A \oplus B \oplus C$:** - XOR is associative, so $D = (A \oplus B) \oplus C$ - Truth table or expression: $$D = A B' C' + A' B C' + A' B' C + A B C$$ 4. **Simplify $E = A'BC + AB'C$:** - Factor $C$: $$E = C(A'B + AB')$$ - Note $A'B + AB' = A \oplus B$ - So, $$E = C (A \oplus B)$$ 5. **Simplify $F = ABC' + (A' + B')C$:** - Distribute: $$F = ABC' + A'C + B'C$$ - This is already in sum of products form. 6. **Simplify $G = ABC$:** - This is a simple AND of all three variables. 7. **Analyze minterm functions $F_1, F_2, F_3$:** - $F_1 = \Sigma(1,4,6)$ means function is 1 at minterms 1,4,6. - $F_2 = \Sigma(3,5)$ means function is 1 at minterms 3,5. - $F_3 = \Sigma(2,4,6,7)$ means function is 1 at minterms 2,4,6,7. 8. **Analyze $F_1(A,B,C,D) = \Sigma(1,2,5,7,8,10,11,13,15)$:** - This is a 4-variable function with minterms listed. **Summary:** - $D$ is the XOR of $A,B,C$. - $E$ is $C$ AND the XOR of $A$ and $B$. - $F$ is sum of products $ABC' + A'C + B'C$. - $G$ is the AND of $A,B,C$. - $F_1, F_2, F_3$ are Boolean functions defined by their minterms. **Final expressions:** $$D = A \oplus B \oplus C$$ $$E = C (A \oplus B)$$ $$F = ABC' + A'C + B'C$$ $$G = ABC$$ These are the simplified and analyzed Boolean functions based on the given expressions and minterms.