Let's solve the given problems step-by-step. --- ## (a) \( F(w, x, y, z) = \Sigma(1, 5, 7, 11, 12, 14, 15) \) ### **Step 1: Find the Complement Function** The complement of \( F \), denoted as \( F' \), contains all minterms not present in \( F \). - Since there are 4 variables, the total number of minterms = \(2^4 = 16\) (from 0 to 15). - Given minterms: \(1, 5, 7, 11, 12, 14, 15\) - Minterms not present: \(0, 2, 3, 4, 6, 8, 9, 10, 13\) So, \[ F'(w, x, y, z) = \Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13) \] ### **Step 2: Express Complement in Maxterm Form** Recall: - **Sum of minterms** (\(\Sigma\)) = **Product of maxterms** (\(\Pi\)) for the complement. So, \[ F'(w, x, y, z) = \Pi(1, 5, 7, 11, 12, 14, 15) \] --- ## (b) \( F(x, y, z) = \Pi(2, 4, 5) \) ### **Step 1: Find the Complement Function** The complement of a function in the maxterm form is the sum of minterms corresponding to the listed maxterms. - 3 variables: minterm numbers from 0 to 7. So, \[ F'(x, y, z) = \Sigma(2, 4, 5) \] ### **Step 2: Express Complement in Maxterm Form** The maxterms not present are for indices: \(0, 1, 3, 6, 7\). So, \[ F'(x, y, z) = \Pi(0, 1, 3, 6, 7) \] --- ## **Final Answers** ### (a) - **Sum-of-minterms:** \(\Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13)\) - **Product-of-maxterms:** \(\Pi(1, 5, 7, 11, 12, 14, 15)\) ### (b) - **Sum-of-minterms:** \(\Sigma(2, 4, 5)\) - **Product-of-maxterms:** \(\Pi(0, 1, 3, 6, 7)\) --- **Let me know if you need the explicit Boolean expressions or further explanation!**

Let's solve each part thoroughly, including detailed steps and calculations. --- ## Part (a) Given: \[ F(w, x, y, z) = \Sigma(1, 5, 7, 11, 12, 14, 15) \] ### **Step 1: Find the complement \(F'\)** - Total number of minterms for 4 variables (\(w, x, y, z\)): \(2^4 = 16\) - \(F\) is 1 at minterms: \(1, 5, 7, 11, 12, 14, 15\) - Therefore, \(F'\) (the complement of \(F\)) will be 1 at all other minterms: \[ \text{Minterms not in }F: 0, 2, 3, 4, 6, 8, 9, 10, 13 \] Thus: \[ F'(w, x, y, z) = \Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13) \] --- ### **Step 2: Express \(F'\) in Product of Maxterms** - Since \(F' = \Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13)\) - The maxterms correspond to the minterms where \(F'\) is 0. - To express \(F'\) in product of maxterms, we take all the maxterms that correspond to the minterms *not* in \(F'\): \[ \text{Maxterms for }F' = \Pi(\text{indices where }F' = 0) = \Pi(1, 5, 7, 11, 12, 14, 15) \] **Hence:** \[ \boxed{ F' = \Pi(1, 5, 7, 11, 12, 14, 15) } \] --- ## Part (b) Given: \[ F(x, y, z) = \Pi(2, 4, 5) \] ### **Step 1: Find the complement \(F'\)** - Since the function is given in product of maxterms, the minterms where \(F\) is 0 are the minterms *not* in the maxterm list. - For 3 variables (\(x, y, z\)), total minterms: \(2^3=8\), i.e., 0 to 7. - Maxterms given: \(2, 4, 5\). These maxterms are 0-based indices. - Minterms where \(F\) is **0**: all minterms *not* in \(\{2, 4, 5\}\): \[ 0, 1, 3, 6, 7 \] **Thus:** \[ F'(x, y, z) = \Sigma(0, 1, 3, 6, 7) \] ### **Step 2: Express \(F'\) as sum of minterms** - \(F' = \Sigma(0, 1, 3, 6, 7)\) - Alternatively, if wanted, we can write the product of maxterms for \(F'\): \[ F' = \Pi(2, 4, 5) \] which is the complement of the original \(F\). --- ## **Final detailed answers:** | Part | Sum of Minterms (SOM) | Product of Maxterms (POM) | |---------|------------------------------|------------------------------| | **(a)** | \(\Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13)\) | \(\Pi(1, 5, 7, 11, 12, 14, 15)\) | | **(b)** | \(\Sigma(0, 1, 3, 6, 7)\) | \(\Pi(2, 4, 5)\) | --- ## Additional notes: - **For (a):** The complement \(F'\) is expressed as a sum of minterms or a product of maxterms depending on the form. We found the minterms where \(F'\) is 1, then expressed its complement in maxterm form. - **For (b):** Given in maxterm form, its complement is naturally expressed as a sum of minterms. --- **If you'd like, I can also derive the canonical Boolean expressions from these minterm and maxterm lists.**

Certainly! Here are the final answers: **(a)** - Sum of minterms (SOM): \(\Sigma(0, 2, 3, 4, 6, 8, 9, 10, 13)\) - Product of maxterms (POM): \(\Pi(1, 5, 7, 11, 12, 14, 15)\) **(b)** - Sum of minterms (SOM): \(\Sigma(0, 1, 3, 6, 7)\) - Product of maxterms (POM): \(\Pi(2, 4, 5)\)