Question

How many select lines would be required for an 8-line-to-1-line multiplexer?

• A. 2
• B. 4
• C. 8
• D. 3

Hint:

Keep these standard Multiplexer configurations in mind: - 2-to-1 MUX: Requires $\log_2(2) = 1$ select line. - 4-to-1 MUX: Requires $\log_2(4) = 2$ select lines. - 8-to-1 MUX: Requires $\log_2(8) = 3$ select lines. - 16-to-1 MUX: Requires $\log_2(16) = 4$ select lines.

Answer 1

The Correct Option is D

Concept: A multiplexer (MUX) is a digital switching circuit that routes one of several input lines to a single common output line. The choice of which input line is connected to the output is controlled by a set of binary select inputs. Mathematically, if a multiplexer has $N$ distinct input lines, the number of required select lines ($m$) must satisfy the exponential routing rule: \[ N = 2^m \] where $m$ is the total number of binary select control bits.

Step 1: Setting up the exponential equation
The problem specifies an 8-line-to-1-line multiplexer. This means:
• Number of data input lines: $N = 8$
• Number of output lines: $1$ Using the relationship formula, we substitute $N = 8$: \[ 8 = 2^m \]

Step 2: Solving for $m$ using base-2 logarithms
To isolate the exponent variable $m$, rewrite $8$ as a power of $2$: \[ 2^3 = 2^m \] Since the bases are identical, their exponents must be equal: \[ m = 3 \] Thus, exactly 3 select lines are required to individually address and route any of the 8 input lines.

Step 3: Verification via Truth Table Options
With $3$ binary select lines ($S_2, S_1, S_0$), the system can generate $2^3 = 8$ unique combinations:
• $000_2 \rightarrow$ Selects Input $I_0$
• $001_2 \rightarrow$ Selects Input $I_1$
• $010_2 \rightarrow$ Selects Input $I_2$
• $011_2 \rightarrow$ Selects Input $I_3$
• $100_2 \rightarrow$ Selects Input $I_4$
• $101_2 \rightarrow$ Selects Input $I_5$
• $110_2 \rightarrow$ Selects Input $I_6$
• $111_2 \rightarrow$ Selects Input $I_7$ This maps perfectly to the $8$ inputs, confirming that fewer lines would not cover all inputs, and more lines would be redundant.