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<div class="container">
<div class="header">
<h1>Digital Logic Design - Final Exam</h1>
<p>Complete Answer Sheet for Uttara University</p>
</div>
<div class="content">
<!-- Question 1: Logic Gates -->
<div class="question-block">
<div class="question">
<span class="q-number">1</span>
<span class="q-text">Logic Gates & Universal Gates</span>
</div>
<div class="answer">
<h3>Q1(a): Define Universal Gates</h3>
<div class="definition">
<p><strong>Universal Gates</strong> are logic gates that can be used to implement any Boolean function without needing any other gate type. The two main universal gates are NAND and NOR.</p>
</div>
<div class="formula">
<strong>NAND:</strong> Y = (A · B)' <br>
<strong>NOR:</strong> Y = (A + B)'
</div>
<h3>Q1(b): Implement Basic Gates using NOR</h3>
<div class="diagram-container">
<img src="./nor_gate.jpg" alt="NOT from NOR" class="diagram-image">
</div>
<div class="sub-section">
<h4>NOT Gate using NOR:</h4>
<p>Connect both inputs of a NOR gate together.</p>
<div class="formula">Y = A ↓ A = (A + A)' = A'</div>
<div class="diagram-container">
<div class="diagram-title">NOT Gate from NOR</div>
<img src="./nor_to_not.jpg" alt="NOT from NOR" class="diagram-image">
</div>
</div>
<div class="sub-section">
<h4>OR Gate using NOR:</h4>
<p>Invert the output of a NOR gate to get an OR function.</p>
<div class="formula">Y = (A ↓ B)' = (A + B)'' = A + B</div>
<div class="diagram-container">
<div class="diagram-title">OR Gate from NOR</div>
<img src="./nor_to_or.jpg" alt="OR from NOR" class="diagram-image">
</div>
</div>
<div class="sub-section">
<h4>AND Gate using NOR:</h4>
<p>Invert the inputs before feeding them into a NOR gate.</p>
<div class="formula">Y = (A' ↓ B') = (A' + B')' = A · B</div>
<div class="diagram-container">
<div class="diagram-title">AND Gate from NOR</div>
<img src="./nor_to_and.jpg" alt="AND from NOR" class="diagram-image">
</div>
</div>
</div>
</div>
<!-- Question 2: K-Map -->
<div class="question-block">
<div class="question">
<span class="q-number">2</span>
<span class="q-text">Karnaugh Map (K-Map) Simplification</span>
</div>
<div class="answer">
<h3>Q2(a): Simplify a 3-bit expression using K-Map</h3>
<div class="definition">
<p>A <strong>Karnaugh Map (K-Map)</strong> is a graphical method to simplify Boolean algebra expressions. It helps minimize the number of logic gates required.</p>
</div>
<p><strong>Example:</strong> Simplify F(A,B,C) = Σm(0,2,4,6)</p>
<div class="diagram-container">
<div class="diagram-title">3-Variable K-Map Example</div>
<img src="https://pantech.ai/wp-content/uploads/2024/09/3var_kmap.png" alt="3-Variable K-Map Example" class="diagram-image">
</div>
<div class="formula">
<strong>Grouping:</strong> Group of four '1's in first and last columns (A'C' and AC').<br>
<strong>Simplified Expression:</strong> F = C'
</div>
<h3>Q2(b): Simplify a 2-bit expression using K-Map</h3>
<p><strong>Example:</strong> Simplify F(A,B) = Σm(1,2)</p>
<div class="diagram-container">
<div class="diagram-title">2-Variable K-Map Example</div>
<img src="https://pantech.ai/wp-content/uploads/2024/09/K-Map-2-var-1-1024x576.png" alt="2-Variable K-Map Example" class="diagram-image">
</div>
<div class="formula">
<strong>Grouping:</strong> Group of two '1's in first row (A'B' and A'B) and single '1' in second row (AB').<br>
<strong>Simplified Expression:</strong> F = A' + B'
</div>
</div>
</div>
<!-- Question 3: Adders and Subtractors -->
<div class="question-block">
<div class="question">
<span class="q-number">3</span>
<span class="q-text">Adders and Subtractors</span>
</div>
<div class="answer">
<h3>Q3(a): Half Adder</h3>
<div class="definition">
<p>A <strong>Half Adder</strong> adds two single bits and produces a Sum and a Carry output. It does not account for a carry-in from a previous stage.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">Half Adder Circuit</div>
<img src="https://media.geeksforgeeks.org/wp-content/uploads/20240404131356/Half-Adder-2.png" alt="Half Adder Circuit" class="diagram-image">
</div>
<div class="formula">
<strong>Boolean Expressions:</strong><br>
Sum (S) = A ⊕ B<br>
Carry (C) = A · B
</div>
<h3>Q3(b): Full Adder using Half Adders</h3>
<div class="definition">
<p>A <strong>Full Adder</strong> adds three bits (A, B, and a Carry-In) and produces a Sum and a Carry-Out. It is the fundamental building block for multi-bit adders.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">Full Adder Circuit</div>
<img src="https://theorycircuit.com/wp-content/uploads/2018/07/full-adder-circuit.png" alt="Full Adder Circuit" class="diagram-image">
</div>
<div class="formula">
<strong>Boolean Expressions:</strong><br>
Sum (S) = A ⊕ B ⊕ Cin<br>
Carry (Cout) = A·B + B·Cin + A·Cin
</div>
<h3>Q3(b.1): Full Adder using Decoder</h3>
<div class="definition">
<p>A <strong>Full Adder</strong> adds three bits (A, B, and a Carry-In) and produces a Sum and a Carry-Out. It is the fundamental building block for multi-bit adders.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">Full Adder Circuit</div>
<img src="https://www.poriyaan.in/media/imgPori/images12/TY3oo29.png" alt="Full Adder Circuit" class="diagram-image">
</div>
<h3>Q3(c): Subtractor</h3>
<div class="definition">
<p>A <strong>Subtractor</strong> is a digital circuit that performs subtraction of binary numbers. There are two types: Half Subtractor (subtracts two bits) and Full Subtractor (subtracts three bits including a borrow).</p>
</div>
<div class="sub-section">
<h4>Half Subtractor</h4>
<div class="formula">
<strong>Boolean Expressions:</strong><br>
Difference (D) = A ⊕ B<br>
Borrow (Bout) = A' · B
</div>
<div class="diagram-container">
<div class="diagram-title">Half Subtractor Circuit</div>
<img src="https://www.electronics-tutorial.net/wp-content/uploads/2015/09/half_sub.png" alt="Half Subtractor Circuit" class="diagram-image">
</div>
</div>
</div>
</div>
<!-- Question 4: Encoders and Decoders -->
<div class="question-block">
<div class="question">
<span class="q-number">4</span>
<span class="q-text">Encoders and Decoders</span>
</div>
<div class="answer">
<h3>Q4(a): Encoder Definition</h3>
<div class="definition">
<p>An <strong>Encoder</strong> is a combinational circuit that converts information from 2^n input lines into an n-bit binary code. It has more inputs than outputs and produces a binary code corresponding to the active input line.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">8-to-3 Encoder</div>
<img src="./8_to_3_encoder.jpg" alt="8-to-3 Encoder" class="diagram-image">
</div>
<h3>Q4(b): Decoder Definition</h3>
<div class="definition">
<p>A <strong>Decoder</strong> is a combinational circuit that converts binary information from n input lines to a maximum of 2^n unique output lines. It has more outputs than inputs and activates a specific output line based on the binary input.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">2-to-4 Decoder</div>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Decoder_Example.svg/500px-Decoder_Example.svg.png" alt="2-to-4 Decoder" class="diagram-image">
</div>
<h3>Q4(c): Encoder vs. Decoder</h3>
<table class="comparison-table">
<tr>
<th>Feature</th>
<th>Encoder</th>
<th>Decoder</th>
</tr>
<tr>
<td><strong>Function</strong></td>
<td>Converts 2^n inputs into an n-bit binary code.</td>
<td>Converts an n-bit binary code into 2^n unique outputs.</td>
</tr>
<tr>
<td><strong>Operation</strong></td>
<td>Compresses data (many-to-few).</td>
<td>Expands data (few-to-many).</td>
</tr>
<tr>
<td><strong>Example</strong></td>
<td>8-to-3 Encoder (8 inputs, 3 outputs).</td>
<td>3-to-8 Decoder (3 inputs, 8 outputs).</td>
</tr>
<tr>
<td><strong>Use Case</strong></td>
<td>Keyboard encoding, converting keypad presses to a binary value.</td>
<td>Memory address decoding, selecting one of many memory locations.</td>
</tr>
</table>
<h3>Q4(d): Applications of Encoder and Decoder</h3>
<div class="applications">
<h4>Applications of Encoder:</h4>
<ul>
<li>Keyboard encoding in computers</li>
<li>Data compression systems</li>
<li>Position encoding in robotics</li>
<li>Analog-to-digital conversion</li>
</ul>
</div>
<div class="applications">
<h4>Applications of Decoder:</h4>
<ul>
<li>Memory address decoding in computer systems</li>
<li>Display drivers (e.g., 7-segment display)</li>
<li>Instruction decoding in CPUs</li>
<li>Data routing in communication systems</li>
</ul>
</div>
</div>
</div>
<!-- Question 5: Multiplexers -->
<div class="question-block">
<div class="question">
<span class="q-number">5</span>
<span class="q-text">Multiplexers (MUX) and Demultiplexers (DEMUX)</span>
</div>
<div class="answer">
<h3>Q5(a): Multiplexer (MUX)</h3>
<div class="definition">
<p>A <strong>Multiplexer (MUX)</strong>, or data selector, is a circuit that selects one of several input signals and forwards it to a single output line. The selection is controlled by a set of select lines.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">4-to-1 Multiplexer (MUX)</div>
<img src="./4_to_1_mux.jpg" alt="4-to-1 MUX Diagram" class="diagram-image">
</div>
<h3>Q5(b): Demultiplexer (DEMUX)</h3>
<div class="definition">
<p>A <strong>Demultiplexer (DEMUX)</strong> performs the reverse operation of a MUX. It takes a single data input and routes it to one of many possible outputs, based on select lines.</p>
</div>
<div class="diagram-container">
<div class="diagram-title">1-to-4 Demultiplexer (DEMUX)</div>
<img src="https://www.electronics-lab.com/wp-content/uploads/2022/10/1_to_4_nand-768x555.png" alt="1-to-4 DEMUX Diagram" class="diagram-image">
</div>
<div class="defination">MUX vs DEMUX — 5 exam-ready points
Definition
MUX: Selects one input from multiple inputs and sends it to a single output.
DEMUX: Takes a single input and routes it to one of many outputs.
Data Direction
MUX: Many → One
DEMUX: One → Many
Select Lines
Both use select lines, but
MUX: Selects which input is connected
DEMUX: Selects which output receives data
Purpose
MUX: Reduces hardware, wiring, and channel usage
DEMUX: Distributes data or control signals to multiple destinations
Use Cases
MUX: Sensor data selection, CPU bus control, communication transmitters
DEMUX: Memory addressing, network packet routing, LED/display control</div>
<h3>Q5(c): Applications of MUX & DEMUX</h3>
<div class="applications">
<h4>Applications of MUX & DEMUX:</h4>
<ul>
<li>Communication systems</li>
<li>Data routing</li>
<li>Control systems</li>
<li>Switching circuits</li>
</ul>
</div>
<div class="highlight">
<h4>Exam Tip:</h4>
<p>Remember that MUX and DEMUX are complementary devices. MUX selects one input from many, while DEMUX routes one input to one of many outputs. Both are controlled by select lines.</p>
</div>
</div>
</div>
</div>
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