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<!--replace-end-7--><!--replace-end-4--><!--replace-end-1--></head><body><div class="ui fluid container universe"><!--replace-start-2--><!--replace-start-3--><!--replace-start-6--><div class="ui text container" id="zettel-container" style="position: relative"><div class="zettel-view"><article class="ui raised attached segment zettel-content"><div class="pandoc"><h1 id="title-h1">Boolean algebra</h1><h2 id="algebraic-laws">Algebraic laws</h2><p>Many of the laws that obtain in the mathematical realm of algebra also obtain for Boolean expressions.</p><h3 id="the-commutative-law">The Commutative Law</h3><p><span class="math display">$$
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x \land y = y \land x \\
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$$</span></p><p>$$</p><pre><code class="language-none">x \lor y = y \lor x</code></pre><p>$$</p><p>Compare the <a href="Whole_numbers.md#the-commutative-property">Commutative Law</a> in the context of arithmetic.</p><h3 id="the-associative-law">The Associative Law</h3><p><span class="math display">$$
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x \land (y \land z) = (x \land y) \land z
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$$</span></p><p><span class="math display">$$
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x \lor (y \lor z) = (x \lor y) \lor z
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$$</span></p><p>Compare the <a href="Whole_numbers.md#the-associative-property">Associative Law</a> in the context of arithmetic.</p><h3 id="the-distributive-law">The Distributive Law</h3><p><span class="math display">$$
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x \land (y \lor z) = (x \land y) \lor (x \land z)
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$$</span></p><p><span class="math display">$$
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x \lor (y \land z) = (x \lor y) \land (x \lor z)
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$$</span></p><p>Compare how the <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: The Distributive Property of Multiplication"><a href="Distributivity.html">Distributive Law applies in the case of algebra based on arithmetic</a></span></span>:</p><p><span class="math display">$$
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a \cdot (b + c) = a \cdot b + a \cdot c
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$$</span></p><h3 id="double-negation-elimination">Double Negation Elimination</h3><p><span class="math display">$$
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\lnot \lnot x = x
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$$</span></p><h3 id="idempotent-law">Idempotent Law</h3><p><span class="math display">$$
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x \land x = x
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$$</span></p><blockquote><p>Combining a quantity with itself either by logical addition or logical multiplication will result in a logical sum or product that is the equivalent of the quantity</p></blockquote><h3 id="demorgans-laws">DeMorgan’s Laws</h3><p>In addition we have <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: DeMorgan’s Laws"><a href="DeMorgan's_Laws.html">DeMorgan’s Laws</a></span></span> which express the relationship that obtains between the negations of conjunctive and disjunctive expressions:</p><p><span class="math display">$$
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\lnot(x \land y) = \lnot x \lor \lnot y
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$$</span></p><p><span class="math display">$$
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\lnot (x \lor y) = \lnot x \land \lnot y
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$$</span></p><h2 id="applying-the-laws-to-simplify-complex-boolean-expressions">Applying the laws to simplify complex Boolean expressions</h2><p>Say we have the following expression:</p><p><span class="math display">$$
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\lnot(\lnot(x) \land \lnot (x \lor y))
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$$</span></p><p>We can employ DeMorgan’s Laws to convert the second conjunct to a different form:</p><p><span class="math display">$$
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\lnot (x \lor y) = \lnot x \land \lnot y
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$$</span></p><p>So now we have:</p><p><span class="math display">$$
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\lnot(\lnot(x) \land (\lnot x \land \lnot y ))
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$$</span></p><p>As we have now have an expression of the form <em>P and (Q and R)</em> we can apply the Distributive Law to simplify the brackets (<em>P and Q and R</em>):</p><p><span class="math display">$$
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\lnot( \lnot(x) \land \lnot(x) \land \lnot(y))
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$$</span></p><p>Notice that we are repeating ourselves in this reformulation. We have <span class="math inline">\(\lnot(x) \land \lnot(x)\)</span> but this is just the same <span class="math inline">\(\lnot(x)\)</span> by the principle of <strong>idempotence</strong>. So we can reduce to:</p><p><span class="math display">$$
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\lnot(\lnot(x) \land \lnot(y))
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$$</span></p><p>This gives our expression the form of the first DeMorgan Law (<span class="math inline">\(\lnot (P \land Q)\)</span>), thus we can apply the law (<span class="math inline">\(\lnot P \lor \lnot Q\)</span>) to get:</p><p><span class="math display">$$
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\lnot(\lnot(x)) \lor \lnot(\lnot(y))
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$$</span></p><p>Of course now we have two double negatives. We can apply the double negation law to get:</p><p><span class="math display">$$
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x \lor y
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$$</span></p><h3 id="truth-table">Truth table</h3><p>Whenever we simplify an algebraic expression the value of the resulting expression should match that of the complex expression. We can demonstrate this with a truth table:</p><table class="ui table"><thead><tr><th><span class="math inline">\(x\)</span></th><th><span class="math inline">\(y\)</span></th><th><span class="math inline">\(\lnot(\lnot(x) \land \lnot (x \lor y))\)</span></th><th><span class="math inline">\(x \lor y\)</span></th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>0</td><td>1</td><td>1</td><td>1</td></tr><tr><td>1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table><h3 id="significance-for-computer-architecture">Significance for computer architecture</h3><p>The fact that we can take a complex Boolean function and reduce it to a simpler formulation has great significance for the development of computer architectures, specifically <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logic gates"><a href="Logic_gates.html">logic gates</a></span></span>. It would be rather resource intensive and inefficient to create a gate that is representative of the complex function. Whereas the simplified version only requires a single <a href="Logic_gates.md#or-gate">OR gate</a>.</p></div></article><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">algebra</span><span class="ui basic label zettel-tag" title="Tag">logic</span><span class="ui basic label zettel-tag" title="Tag">nand-to-tetris</span><span class="ui basic label zettel-tag" title="Tag">propositional-logic</span></div></nav><nav class="ui bottom attached icon compact inverted menu blue" id="neuron-nav-bar"><!--replace-start-9--><!--replace-end-9--><a class="right item" href="impulse.html" title="Open Impulse"><i class="wave square icon"></i></a></nav></div></div><!--replace-end-6--><!--replace-end-3--><!--replace-end-2--><div class="ui center aligned container footer-version"><div class="ui tiny image"><a href="https://neuron.zettel.page"><img alt="logo" src="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" title="Generated by Neuron 1.9.35.3" /></a></div></div></div></body></html> |