1. **State the problem:** We are given a Boolean function $F(A,B,C,D)$ defined by the minterms $\Sigma(1, 3, 7, 9, 11, 15)$. Our goal is to express this function in a simplified Boolean expression. 2. **Recall the formula and rules:** The function $F$ is true for the minterms listed. Each minterm corresponds to a unique combination of variables where the function outputs 1. 3. **Write minterms in binary:** - 1 = 0001 - 3 = 0011 - 7 = 0111 - 9 = 1001 - 11 = 1011 - 15 = 1111 4. **Express each minterm as a product (AND) of literals:** - $m_1 = \overline{A}\overline{B}\overline{C}D$ - $m_3 = \overline{A}\overline{B}CD$ - $m_7 = \overline{A}BCD$ - $m_9 = A\overline{B}\overline{C}D$ - $m_{11} = A\overline{B}CD$ - $m_{15} = ABCD$ 5. **Sum of minterms expression:** $$F = m_1 + m_3 + m_7 + m_9 + m_{11} + m_{15}$$ 6. **Simplify the expression using Boolean algebra:** Group terms to factor common literals: - Group 1: $m_1 + m_3 = \overline{A}\overline{B}D(\overline{C} + C) = \overline{A}\overline{B}D$ - Group 2: $m_7 = \overline{A}BCD$ - Group 3: $m_9 + m_{11} = A\overline{B}D(\overline{C} + C) = A\overline{B}D$ - Group 4: $m_{15} = ABCD$ 7. **Combine groups:** $$F = \overline{A}\overline{B}D + \overline{A}BCD + A\overline{B}D + ABCD$$ 8. **Factor further:** - Factor $D$ from terms with $D$: $$F = D(\overline{A}\overline{B} + A\overline{B}) + BCD(\overline{A} + A)$$ - Since $\overline{A} + A = 1$: $$F = D(\overline{B}(\overline{A} + A)) + BCD = D\overline{B} + BCD$$ 9. **Final simplified expression:** $$F = D\overline{B} + BCD$$ This is the minimal sum of products form for the given function.