Let's break down the solution for the given Boolean function step by step for each device. --- ## **Given Function** \[ f(a,b,c,d) = \prod M(1, 5, 7, 9, 10, 11, 12) \] This means \( f \) is **0** for minterms 1, 5, 7, 9, 10, 11, 12 and **1** elsewhere. --- # **(a) Using 16:1 MUX** ### **Step-by-step Solution** 1. **Inputs:** - There are 4 variables: \( a, b, c, d \). They can address all 16 combinations (0 to 15). 2. **Connection:** - Use \( a, b, c, d \) as the select lines \( S_3, S_2, S_1, S_0 \) of the MUX. - The 16 data inputs (\( I_0 \) to \( I_{15} \)) should be connected to logic '1' or '0' based on the function’s output for each minterm. 3. **Assign Data Inputs:** - For minterms 1, 5, 7, 9, 10, 11, 12: Output is 0. - Rest: Output is 1. | Input | Output | |-------|--------| | 0 | 1 | | 1 | 0 | | 2 | 1 | | 3 | 1 | | 4 | 1 | | 5 | 0 | | 6 | 1 | | 7 | 0 | | 8 | 1 | | 9 | 0 | | 10 | 0 | | 11 | 0 | | 12 | 0 | | 13 | 1 | | 14 | 1 | | 15 | 1 | 4. **Final Answer:** - Connect \( a, b, c, d \) as select lines. - Connect data lines as above. --- # **(b) Using 8:1 MUX** ### **Step-by-step Solution** 1. **Inputs:** 4 variables, but 8:1 MUX can select from only 8 inputs. 2. **Method:** Use 3 variables as select lines, and express the function in terms of the fourth variable. - Let \( c, d, b \) be select lines. \( a \) will be used in data input expressions. 3. **Find Data Inputs:** Each data input will be an expression in \( a \). - Data inputs \( I_0 \) to \( I_7 \) correspond to all combinations of \( c, d, b \). - For each, plug in values for \( c, d, b \) and see for which values of \( a \) the output is 0 or 1. > This step is a bit lengthy. If you want the explicit expressions for each data input, let me know! --- # **(c) Using 4:1 MUX** ### **Step-by-step Solution** 1. **Inputs:** 4 variables, 4:1 MUX. 2. **Method:** Use 2 variables as select lines, express function in terms of the other 2. - Let’s use \( c, d \) as select lines, and \( a, b \) for the data inputs. 3. **For each combination of \( c, d \) (00, 01, 10, 11):** - Write the function in terms of \( a, b \) and connect the corresponding logic to each data input. --- # **(d) Using 1:16 DEMUX** ### **Step-by-step Solution** 1. **DEMUX takes 1 input and routes it to one of 16 outputs.** 2. **Use \( a, b, c, d \) as select lines.** 3. **Connect the 1 input to logic HIGH (1).** 4. **The required function is 1 except for minterms 1, 5, 7, 9, 10, 11, 12.** 5. **OR all outputs except these minterms** (meaning: output is high unless selected minterm). --- # **(e) Using 4 x 16 Decoder** ### **Step-by-step Solution** 1. **Decoder gives 16 outputs, each corresponding to one minterm.** 2. **OR together all outputs except 1, 5, 7, 9, 10, 11, 12**. - \( f = Y_0 + Y_2 + Y_3 + Y_4 + Y_6 + Y_8 + Y_{13} + Y_{14} + Y_{15} \) - Where \( Y_i \) is the output for minterm \( i \). --- ## **Summary Table** | Device | Variables as Select Lines | Data Inputs / Logic | |---------------|--------------------------|-----------------------------| | 16:1 MUX | a, b, c, d | As per minterm table above | | 8:1 MUX | 3 variables | Each input: expr in 4th var | | 4:1 MUX | 2 variables | Each input: expr in 2 vars | | 1:16 DEMUX | a, b, c, d | OR all except 1,5,7,9,10,11,12 | | 4x16 Decoder | a, b, c, d | OR all except 1,5,7,9,10,11,12 | --- ## **Final Answer Example (16:1 MUX):** \[ \boxed{ \text{Connect } a, b, c, d \text{ as select lines to 16:1 MUX.} \\ \text{Set inputs:} \ I_0=1, I_1=0, I_2=1, I_3=1, I_4=1, I_5=0, I_6=1, I_7=0, I_8=1, I_9=0, I_{10}=0, I_{11}=0, I_{12}=0, I_{13}=1, I_{14}=1, I_{15}=1 } \] --- Let me know if you want the detailed expressions for the 8:1 or 4:1 MUX cases!