Final answer:
To design the function OUT(X,Y,Z) using a Multiplexer, one can use an 8-to-1 MUX with select lines connected to the inputs X, Y, and Z. Data inputs for the three minterms that produce '1' should be set to '1', while the rest can be set to '0' or optimized using don't care conditions.
Step-by-step explanation:
The function specified in the question is OUT(X,Y,Z) = X′ ⋅ Y ⋅ Z + X ⋅ Y ⋅ Z′ + X′ ⋅ Y′ ⋅ Z′. To design this function using a Multiplexer (MUX), we would typically use an 8-to-1 MUX because the function depends on three variables (X, Y, Z), which would generate 2^3 = 8 different combinations. However, since the function only involves 3 unique minterms, we can use a smaller MUX and additional logic gates if necessary.
First, let's assign each minterm to a corresponding input line of the MUX. For an 8-to-1 MUX, we have the inputs I0 to I7, and the select lines would be S0 (Z), S1 (Y), and S2 (X). The function is 1 when:
- XYZ = 100, corresponding to I4: X′ ⋅ Y ⋅ Z
- XYZ = 011, corresponding to I3: X ⋅ Y ⋅ Z′
- XYZ = 000, corresponding to I0: X′ ⋅ Y′ ⋅ Z′
Since the minterms don't include all possible input lines, we can use don't care terms for the other inputs to simplify the design. Connect the data inputs I4, I3, and I0 to '1', and set the rest as '0' or connect them to either '0' or '1' if you optimize with don't care conditions.
The select lines S0, S1, and S2 of the MUX will be directly connected to Z, Y, and X respectively. Thus, when a specific combination of inputs is selected, it will output '1' for our function's truth conditions, and '0' otherwise, providing the desired output.