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chapter2.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link
rel="stylesheet"
href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/bootstrap.min.css"
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crossorigin="anonymous"
/>
<title>Chapter-2</title>
</head>
<body>
<nav class="navbar navbar-expand-lg navbar-dark bg-dark">
<a class="navbar-brand" href="index.html">Digital Logic</a>
<button
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarTogglerDemo02"
aria-controls="navbarTogglerDemo02"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="collapse navbar-collapse" id="navbarTogglerDemo02">
<ul class="navbar-nav mr-auto mt-2 mt-lg-0">
<li class="nav-item">
<a class="nav-link" href="chapter1.html">CHAPTER-1</a>
</li>
<li class="nav-item">
<a class="nav-link active">CHAPTER-2</a>
</li>
<li class="nav-item">
<a class="nav-link" href="chapter3.html">CHAPTER-3</a>
</li>
<li class="nav-item">
<a class="nav-link" href="chapter4.html">CHAPTER-4</a>
</li>
<li class="nav-item">
<a class="nav-link" href="chapter5.html">CHAPTER-5</a>
</li>
</ul>
</div>
</nav>
<br />
<!-- main-content -->
<div class="col-lg-6 justify-content-center m-auto">
<h5 class="card-header shadow p-3 mb-2 bg-white rounded" id="ch2_t1">
Postulates and Theorems of Boolean Algebra
</h5>
<div class="card m-3">
<div class="card-body">
<table class="table table-striped">
<tbody>
<tr>
<th scope="row">Postulate 2</th>
<td>x+0 = x</td>
<td>x⋅1 = x</td>
</tr>
<tr>
<th scope="row">Postulate 5</th>
<td>x+x′ = 1</td>
<td>x⋅x′ = 0</td>
</tr>
<tr>
<th scope="row">Theorem 1</th>
<td>x+x = x</td>
<td>x⋅x = x</td>
</tr>
<tr>
<th scope="row">Theorem 2</th>
<td>x+1 = 1</td>
<td>x⋅0 = 0</td>
</tr>
<tr>
<th scope="row">
Theorem 3,<br />
Involution
</th>
<td>(x′)′ = x</td>
<td></td>
</tr>
<tr>
<th scope="row">Postulate 3,<br />commutative</th>
<td>x+y = y+x</td>
<td>xy = yx</td>
</tr>
<tr>
<th scope="row">Theorem 4,<br />associative</th>
<td>x+(y+z) = (x+y)+z</td>
<td>x(yz) = (xy)z</td>
</tr>
<tr>
<th scope="row">Postulate 4,<br />distributive</th>
<td>x(y+z) = xy+xz</td>
<td>x+yz = (x+y) (x+z)</td>
</tr>
<tr>
<th scope="row">Theorem 5,<br />DeMorgan</th>
<td>(x+y)′ = x′y′</td>
<td>(xy)′ = x′+y′</td>
</tr>
<tr>
<th scope="row">Theorem 6,<br />absorption</th>
<td>x+xy = x</td>
<td>x(x+y) = x</td>
</tr>
</tbody>
</table>
</div>
</div>
<h5 class="card-header shadow p-3 mb-2 bg-white rounded" id="ch2_t2">
Important Proof
</h5>
<div class="card m-3">
<div class="card-body">
<p><strong>1. </strong> A + AB = A</p>
<p><strong>2. </strong> A . (A + B) = A</p>
<p><strong>3. </strong> A + A'B = A + B</p>
<p><strong>4. </strong> A . (A' + B) = AB</p>
<p><strong>5. </strong> AB + BC + A'C = AB + A'C</p>
<p>
<strong>6. </strong> (A + B) (B + C) (A' + C) = (A + B) (A' + C)
</p>
</div>
</div>
<h5 class="card-header shadow p-3 mb-2 bg-white rounded" id="ch2_t3">
Duality
</h5>
<div class="card m-3">
<div class="card-body">
<p>To find duality convert (+) to (.) or vice-versa</p>
<p><strong>Example:</strong> A + B = A . B</p>
</div>
</div>
<h5 class="card-header shadow p-3 mb-2 bg-white rounded" id="ch2_t4">
Complement of a Function
</h5>
<div class="card m-3">
<div class="card-body">
<p><strong>Step 1:</strong> Find the dual of the function.</p>
<p><strong>Step 2:</strong> Complement each literals/varaibles.</p>
<p><strong>Example:</strong> F1 = X'YZ' + X'Y'Z</p>
<p>S1 => F1 = (X' + Y + Z') . (X' + Y' + Z)</p>
<p>S2 => F1' = (X + Y' + Z) . (X + Y + Z') //answer</p>
</div>
</div>
<h5 class="card-header shadow p-3 mb-2 bg-white rounded" id="ch2_t5">
Canonical and Standard Forms
</h5>
<div class="card m-3">
<div class="card-body">
<h6>Sum of minterms</h6>
<p>
A minterm, denoted as m<sub>i</sub>, is a product (AND)
<br />uncomplemented X = 1,<br />complemented X' = 0
</p>
<p>
<strong>Example : </strong><br />F = x + yz <br />= x (y + y') (z +
z') + (x + x') yz <br />= xyz + xyz' + xy' z + xy' z' + xyz + x' yz
<br />= x' yz + xy' z' + xy' z + xyz' + xyz <br />= m<sub>3</sub> +
m<sub>4</sub> + m<sub>5</sub> + m<sub>6</sub> + m<sub>7</sub> <br />
= ∑(3, 4, 5, 6, 7)
</p>
<h6>Product of maxterms</h6>
<p>
A maxterm, denoted as M<sub>i</sub>, is a product (OR)
<br />uncomplemented X = 0,<br />complemented X' = 1
</p>
<p>
<strong>Example : </strong><br />F = (x+y+z) • (x+y+z') • (x+y'+z) •
(x'+y+z) <br />M<sub>0</sub>•M<sub>1</sub>•M<sub>2</sub>•M<sub
>4</sub
>
<br />
= ∏(0, 1, 2, 4)
</p>
<br />
<table class="table table-sm">
<thead class="thead-dark">
<tr>
<th scope="col">x</th>
<th scope="col">y</th>
<th scope="col">z</th>
<th scope="col">Minterms</th>
<th scope="col">Maxterms</th>
<th scope="col">F</th>
<th scope="col">F'</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0</td>
<td>0</td>
<td>m<sub>0</sub>= x'y'z'</td>
<td>M<sub>0</sub>= x+y+z</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<th>0</th>
<td>0</td>
<td>1</td>
<td>m<sub>1</sub>= x'y'z</td>
<td>M<sub>1</sub>= x+y+z'</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<th>0</th>
<td>1</td>
<td>0</td>
<td>m<sub>2</sub>= x'yz'</td>
<td>M<sub>2</sub>= x+y'+z</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<th>0</th>
<td>1</td>
<td>1</td>
<td>m<sub>3</sub>= x'yz</td>
<td>M<sub>3</sub>= x+y'+z'</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<th>1</th>
<td>0</td>
<td>0</td>
<td>m<sub>4</sub>= xy'z'</td>
<td>M<sub>4</sub>= x'+y+z</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<th>1</th>
<td>0</td>
<td>1</td>
<td>m<sub>5</sub>= xy'z</td>
<td>M<sub>5</sub>= x'+y+z'</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>0</td>
<td>m<sub>6</sub>= xyz'</td>
<td>M<sub>6</sub>= x'+y'+z</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>1</td>
<td>m<sub>7</sub>= xyz</td>
<td>M<sub>7</sub>= x'+y'+z'</td>
<td>0</td>
<td>1</td>
</tr>
</tbody>
</table>
<hr />
<h6>Canonical form and Standard Form</h6>
<ul>
<li>
Canonical form -
<ul>
<li>Sum of minterms (SOM)</li>
<li>Product of maxterms (POM)</li>
</ul>
</li>
<li>
Standard forms (may use less gates) -
<ul>
<li>Sum of products (SOP)</li>
<li>Product of sums (POS)</li>
</ul>
</li>
<li>SOP form may not be in Canonical Form</li>
</ul>
<div class="card">
<ul class="list-group list-group-flush">
<li class="list-group-item">
F = ab+a’ (already<strong> sum of products:SOP</strong>)<br />F
= ab + a’(b+b’) (expanding term)<br />F = ab + a’b + a’b’ (it is
<strong>canonical form:SOM</strong>)
</li>
</ul>
</div>
</div>
</div>
</div>
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