1. **Problem a:** Find the largest decimal value that can be represented using 12 bits. 2. **Solution a:** The largest decimal number in binary with $n$ bits is when all bits are 1. $$\text{Largest value} = 2^{12} - 1 = 4096 - 1 = 4095$$ So, the largest decimal value is **4095**. 3. **Problem b:** Determine and simplify the output expression for the circuit with inputs A, B, and C, where A and B go to an AND gate, C goes through a NOT gate, and the output is the AND of A, B, and $\overline{C}$. 4. **Solution b:** The output expression is: $$F = A \cdot B \cdot \overline{C}$$ This is already simplified. 5. **Problem c:** What is universal logic? Implement XOR gate using NAND gates. 6. **Solution c:** Universal logic means a set of gates that can implement any Boolean function. NAND gates alone are universal. To implement XOR using NAND gates: $$XOR(A,B) = (A \text{ NAND } (A \text{ NAND } B)) \text{ NAND } (B \text{ NAND } (A \text{ NAND } B))$$ This uses only NAND gates to create XOR. 7. **Problem 6(a):** Simplify the Boolean function $F(w,x,y,z) = \Sigma(1,3,7,11,15)$ with don't-care $d(w,x,y,z) = \Sigma(0,2,5)$. 8. **Solution 6(a):** Using Karnaugh map simplification including don't-cares, the simplified function is: $$F = y z + w x y z$$ 9. **Problem 6(b):** Simplify $F(A,B,C,D) = \Sigma(0,1,2,5,8,9,10)$ in product of sums form. 10. **Solution 6(b):** The product of sums form after simplification is: $$F = (A + B + C)(A + \overline{B} + D)(\overline{A} + B + \overline{D})$$ 11. **Problem 6(c):** Use DeMorgan's theorems to convert $y = A + B + CD$ to an expression with only single-variable inversions. 12. **Solution 6(c):** Applying DeMorgan's theorem: $$y = A + B + C D = \overline{\overline{A + B + C D}} = \overline{\overline{A} \cdot \overline{B} \cdot \overline{C D}} = \overline{\overline{A} \cdot \overline{B} \cdot (\overline{C} + \overline{D})}$$ This expression contains only single-variable inversions $\overline{A}$, $\overline{B}$, $\overline{C}$, and $\overline{D}$. 13. **Additional 6(a) alternative:** Simplify $F(w,x,y,z) = \Sigma(0,1,2,4,5,6,8,9,12,13,14)$. Simplified form: $$F = \overline{w} \overline{x} + \overline{y} \overline{z}$$ 14. **Additional 6(b) alternative:** Voting machine with inputs A, B, C; output is 1 if at least two inputs are 1. Boolean expression: $$F = A B + B C + A C$$ Simplified as is. 15. **Additional 6(c) alternative:** EX-OR and EX-NOR gates symbols and truth tables: - EX-OR output is 1 if inputs differ. - EX-NOR output is 1 if inputs are the same. Truth table for EX-OR (A,B): | A | B | Output | |---|---|--------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 | Truth table for EX-NOR (A,B): | A | B | Output | |---|---|--------| | 0 | 0 | 1 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 |