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Question 97

Computer Awareness Computer memory Hard

Consider 4-bit gray code representation of numbers. Let  h<sub>3</sub> h<sub>2</sub> h<sub>1</sub> h<sub>0</sub> be the gray code representation of a number n and g<sub>1</sub>g<sub>2</sub>g<sub>3</sub>g<sub>0</sub> be the gray code representation of the number (n + 1) modulo 16. Which one of the following functions is correct?

(A) g<sub>0</sub> (h<sub>3</sub> h<sub>2</sub> h<sub>1</sub> h<sub>0</sub>) = Σ(1,2,3,6,10,13,14,15)

(B) g<sub>1 </sub>(h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">3</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">2</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">0</span>) = Σ(4,9,10,11,12,13,14,15)

(C) g<sub>2</sub> (h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">3</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">2</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">1</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">0</span>) = Σ(2,4,5,6,7,12,13,15)

(D) g<sub>3</sub> (h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">3</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">2</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">1</span> h<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">0</span>) = Σ(0,1,6,7,10,11,12,13)

✓ Correct Answer: Option C Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

Build the table: for each 4-bit Gray code h (the code of n), find the Gray code of \((n+1) \bmod 16\) and note which bits are 1.

Gray sequence: 0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000, and then back to 0000.

Working this out bit by bit (h read as a decimal number):

\(g_0 = \Sigma(0,1,6,7,10,11,12,13)\)
\(g_1 = \Sigma(1,2,3,6,10,13,14,15)\)
\(g_2 = \Sigma(2,4,5,6,7,12,13,15)\)
\(g_3 = \Sigma(4,9,10,11,12,13,14,15)\)

Only option (C) is correct, \(g_2 = \Sigma(2,4,5,6,7,12,13,15)\). The other options pair a correct list with the wrong output bit.

Answer: (C)

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