1. **Stating the problem:** We need to solve for the function $$F_3 = \Sigma(2, 4, 6, 7)$$ where the variables are $A, B, C, D$. 2. **Understanding the problem:** The notation $$\Sigma(m_1, m_2, \ldots)$$ represents the sum of minterms for the given indices. Each minterm corresponds to a binary combination of the variables $A, B, C, D$ where the function is 1. 3. **Variables and minterms:** Since $F_3$ is a function of four variables $A, B, C, D$, each minterm index corresponds to a 4-bit binary number representing $ABCD$ in order. 4. **Convert minterm indices to binary:** - $2_{10} = 0010_2$ corresponds to $A=0, B=0, C=1, D=0$ - $4_{10} = 0100_2$ corresponds to $A=0, B=1, C=0, D=0$ - $6_{10} = 0110_2$ corresponds to $A=0, B=1, C=1, D=0$ - $7_{10} = 0111_2$ corresponds to $A=0, B=1, C=1, D=1$ 5. **Write minterms in Boolean form:** - For $2$: $A' B' C D'$ - For $4$: $A' B C' D'$ - For $6$: $A' B C D'$ - For $7$: $A' B C D$ 6. **Sum of minterms expression:** $$F_3 = A' B' C D' + A' B C' D' + A' B C D' + A' B C D$$ 7. **Simplify the expression:** Group terms with common factors: - Factor $A'$ out: $$F_3 = A' (B' C D' + B C' D' + B C D' + B C D)$$ 8. **Simplify inside the parentheses:** Group terms with $D'$: $$B' C D' + B C' D' + B C D' = D' (B' C + B C' + B C)$$ Note that $B' C + B C' + B C = C (B' + B) + B C' = C (1) + B C' = C + B C'$ Since $C + B C' = C + B C' = (C + B)(C + C') = (C + B)(1) = B + C$ So, $$D' (B' C + B C' + B C) = D' (B + C)$$ The remaining term is $B C D$. So inside parentheses: $$D' (B + C) + B C D$$ 9. **Final simplified expression:** $$F_3 = A' [D' (B + C) + B C D]$$ 10. **Interpretation:** The function $F_3$ is true when $A=0$ and either $D=0$ and $(B=1$ or $C=1)$, or when $B=1, C=1,$ and $D=1$. **Final answer:** $$\boxed{F_3 = A' [D' (B + C) + B C D]}$$