Question

A 4-variable Boolean function is implemented using an 8:1 multiplexer. Three variables are used as select lines, and the remaining variable is connected appropriately to the data inputs. The minimum number of 8:1 multiplexers required to realize any arbitrary 4-variable Boolean function is

• A. \(1 \)
• B. \(2 \)
• C. \(4 \)
• D. \(8 \)

Hint:

General MUX implementation rule of thumb: To implement an arbitrary function of $k$ variables: - You need exactly one multiplexer of size $2^{k-1}:1$. Here, $k = 4$ variables, so we need a single $2^{4-1}:1 = 8:1$ MUX.

Answer 1

The Correct Option is A

Concept: A multiplexer (MUX) acts as a combinational selector switch. An $8:1$ MUX contains $2^3 = 8$ input data lines ($I_0$ to $I_7$) and $3$ select lines ($S_2, S_1, S_0$). According to general multiplexer expansion properties:
• A MUX with $n$ select lines can naturally implement any logic function containing up to $n$ variables by mapping those variables to the select lines and tying individual data inputs to Logic $0$ or Logic $1$.
• Impressively, by using the implementation variable approach, an $n$-select-line MUX can also implement any arbitrary function of $(n+1)$ variables. The extra variable is routed to the data input pins in regular, inverted, or constant formats ($0, 1, x, \bar{x}$).

Step 1: Mapping the 4 variables to the 8:1 MUX architecture.
Let the arbitrary 4-variable Boolean function be $f(A, B, C, D)$. We choose three of these variables, say $A, B,$ and $C$, and tie them directly to the three select lines of our multiplexer: \[ S_2 = A, \quad S_1 = B, \quad S_0 = C \] These three lines can form $2^3 = 8$ distinct binary combinations ($000$ through $111$), which match the 8 input lines ($I_0$ through $I_7$) of our multiplexer.

Step 2: Evaluating the inputs using the remaining variable.
The remaining fourth variable, $D$, along with constants, will be applied to the data input channels. For each distinct combination of the selection variables $A, B, C$, the output function will behave in one of four ways depending on the state of $D$:
• Independent of $D$ and always low: \( I_n = 0 \)
• Independent of $D$ and always high: \( I_n = 1 \)
• Equal to $D$: \( I_n = D \)
• Equal to the complement of $D$: \( I_n = \bar{D} \)

Step 3: Conclusion on hardware count.
Because every single one of the 8 sub-minterm combinations can be uniquely satisfied by applying either $0, 1, D,$ or $\bar{D}$ to the data pins, a single 8:1 multiplexer unit is entirely sufficient to build any arbitrary 4-variable function. No extra multiplexers or combining logic gates are needed. Hence, the correct choice is option (1).