{"id":222289,"date":"2025-05-31T08:03:09","date_gmt":"2025-05-31T08:03:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222289"},"modified":"2025-05-31T08:03:11","modified_gmt":"2025-05-31T08:03:11","slug":"design-a-2-bit-comparator","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/31\/design-a-2-bit-comparator\/","title":{"rendered":"Design a 2-bit comparator"},"content":{"rendered":"\n<p>Design a 2-bit comparator. The comparator input has two 2-bit numbers A and B. A consists of 2 bits a1 and a0. B consists of 2 bits b1 and b0. The comparator has three outputs Z2, Z1, Z0. a) Show the truth table for this comparator b) Write the logic equation for Z2 that are true if only if A &lt; b=&#8221;&#8221; c)=&#8221;&#8221; write=&#8221;&#8221; the=&#8221;&#8221; logic=&#8221;&#8221; equation=&#8221;&#8221; for=&#8221;&#8221; z1=&#8221;&#8221; that=&#8221;&#8221; are=&#8221;&#8221; true=&#8221;&#8221; if=&#8221;&#8221; only=&#8221;&#8221; if=&#8221;&#8221; a=&#8221;&#8221;&gt; B d) Write the logic equation for Z0 that are true if only if A = B e) Show the logic design for this comparator using the minimum number of AND, OR, and Inverters. f) Use Logisim to verify your design<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Problem Setup:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Inputs:<\/strong>\n<ul class=\"wp-block-list\">\n<li><code>A = a1 a0<\/code> (2-bit number)<\/li>\n\n\n\n<li><code>B = b1 b0<\/code> (2-bit number)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Outputs:<\/strong>\n<ul class=\"wp-block-list\">\n<li><code>Z2 = 1<\/code> if <code>A &lt; B<\/code><\/li>\n\n\n\n<li><code>Z1 = 1<\/code> if <code>A > B<\/code><\/li>\n\n\n\n<li><code>Z0 = 1<\/code> if <code>A = B<\/code><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>a) Truth Table<\/strong><\/h3>\n\n\n\n<p>| a1 | a0 | b1 | b0 || A | B | Z2 (<code>A &lt; B<\/code>) | Z1 (<code>A &gt; B<\/code>) | Z0 (<code>A = B<\/code>) |<br>|&#8212;-|&#8212;-|&#8212;-|&#8212;-||&#8212;-|&#8212;-|&#8212;&#8212;&#8212;&#8212;&#8211;|&#8212;&#8212;&#8212;&#8212;&#8211;|&#8212;&#8212;&#8212;&#8212;&#8211;|<br>| 0 | 0 | 0 | 0 || 0 | 0 | 0 | 0 | 1 |<br>| 0 | 0 | 0 | 1 || 0 | 1 | 1 | 0 | 0 |<br>| 0 | 0 | 1 | 0 || 0 | 2 | 1 | 0 | 0 |<br>| 0 | 0 | 1 | 1 || 0 | 3 | 1 | 0 | 0 |<br>| 0 | 1 | 0 | 0 || 1 | 0 | 0 | 1 | 0 |<br>| 0 | 1 | 0 | 1 || 1 | 1 | 0 | 0 | 1 |<br>| 0 | 1 | 1 | 0 || 1 | 2 | 1 | 0 | 0 |<br>| 0 | 1 | 1 | 1 || 1 | 3 | 1 | 0 | 0 |<br>| 1 | 0 | 0 | 0 || 2 | 0 | 0 | 1 | 0 |<br>| 1 | 0 | 0 | 1 || 2 | 1 | 0 | 1 | 0 |<br>| 1 | 0 | 1 | 0 || 2 | 2 | 0 | 0 | 1 |<br>| 1 | 0 | 1 | 1 || 2 | 3 | 1 | 0 | 0 |<br>| 1 | 1 | 0 | 0 || 3 | 0 | 0 | 1 | 0 |<br>| 1 | 1 | 0 | 1 || 3 | 1 | 0 | 1 | 0 |<br>| 1 | 1 | 1 | 0 || 3 | 2 | 0 | 1 | 0 |<br>| 1 | 1 | 1 | 1 || 3 | 3 | 0 | 0 | 1 |<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>b) Logic Equation for Z2 (A &lt; B)<\/strong><\/h3>\n\n\n\n<pre class=\"wp-block-code\"><code>Z2 = (\u00aca1 \u2227 b1) \u2228 (a1 \u2261 b1) \u2227 (\u00aca0 \u2227 b0)\n<\/code><\/pre>\n\n\n\n<p>Using basic logic gates:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Z2 = (\u00aca1 \u2227 b1) \u2228 ((a1 \u2227 b1) \u2228 (\u00aca1 \u2227 \u00acb1)) \u2227 (\u00aca0 \u2227 b0)\n<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>c) Logic Equation for Z1 (A &gt; B)<\/strong><\/h3>\n\n\n\n<pre class=\"wp-block-code\"><code>Z1 = (a1 \u2227 \u00acb1) \u2228 (a1 \u2261 b1) \u2227 (a0 \u2227 \u00acb0)\n<\/code><\/pre>\n\n\n\n<p>Using basic gates:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Z1 = (a1 \u2227 \u00acb1) \u2228 ((a1 \u2227 b1) \u2228 (\u00aca1 \u2227 \u00acb1)) \u2227 (a0 \u2227 \u00acb0)\n<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>d) Logic Equation for Z0 (A = B)<\/strong><\/h3>\n\n\n\n<pre class=\"wp-block-code\"><code>Z0 = (a1 \u2261 b1) \u2227 (a0 \u2261 b0)\n<\/code><\/pre>\n\n\n\n<p>Which simplifies to:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Z0 = (a1 \u2299 b1) \u2227 (a0 \u2299 b0)  \/\/ XNOR gates\n<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>e) Logic Design Using Minimum Gates<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>XNOR Gate:<\/strong> Use 2 for Z0.<\/li>\n\n\n\n<li><strong>AND\/OR Gates:<\/strong> For Z1 and Z2, use a mix of basic AND, OR, NOT gates.<\/li>\n\n\n\n<li>Use gate sharing (e.g., use a1 \u2261 b1 in both Z1 and Z2).<\/li>\n<\/ul>\n\n\n\n<p>Diagram layout (for Logisim or circuit):<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Generate <code>\u00aca1<\/code>, <code>\u00aca0<\/code>, <code>\u00acb1<\/code>, <code>\u00acb0<\/code><\/li>\n\n\n\n<li>Compute <code>a1 \u2261 b1<\/code>, <code>a0 \u2261 b0<\/code><\/li>\n\n\n\n<li>Build <code>Z0 = (a1 \u2261 b1) \u2227 (a0 \u2261 b0)<\/code><\/li>\n\n\n\n<li>Compute:\n<ul class=\"wp-block-list\">\n<li><code>Z2 = (\u00aca1 \u2227 b1) \u2228 ((a1 \u2261 b1) \u2227 (\u00aca0 \u2227 b0))<\/code><\/li>\n\n\n\n<li><code>Z1 = (a1 \u2227 \u00acb1) \u2228 ((a1 \u2261 b1) \u2227 (a0 \u2227 \u00acb0))<\/code><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>f) Use Logisim to Verify<\/strong><\/h3>\n\n\n\n<p>In Logisim:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Create inputs <code>a1<\/code>, <code>a0<\/code>, <code>b1<\/code>, <code>b0<\/code><\/li>\n\n\n\n<li>Use NOT, AND, OR, and XNOR gates as above<\/li>\n\n\n\n<li>Display outputs <code>Z2<\/code>, <code>Z1<\/code>, <code>Z0<\/code><\/li>\n\n\n\n<li>Simulate all 16 input combinations<\/li>\n\n\n\n<li>Confirm results match truth table<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation <\/strong><\/h3>\n\n\n\n<p>A 2-bit comparator checks the relationship between two binary numbers (A and B) and determines whether A is less than, greater than, or equal to B. Each number is represented using two bits (<code>a1a0<\/code> and <code>b1b0<\/code>), giving 16 total input combinations.<\/p>\n\n\n\n<p>We use three outputs:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>Z2 = 1<\/code> if A &lt; B<\/li>\n\n\n\n<li><code>Z1 = 1<\/code> if A > B<\/li>\n\n\n\n<li><code>Z0 = 1<\/code> if A = B<\/li>\n<\/ul>\n\n\n\n<p>The truth table explicitly compares all combinations of A and B from 0 to 3. For each pair, the comparator outputs only one <code>1<\/code> among <code>Z0<\/code>, <code>Z1<\/code>, and <code>Z2<\/code>.<\/p>\n\n\n\n<p>To implement this in logic, we consider bitwise comparisons:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For equality, both bits must be equal. This is easily implemented using XNOR gates.<\/li>\n\n\n\n<li>For greater than or less than, we compare the most significant bits (<code>a1<\/code> vs. <code>b1<\/code>). If they differ, the result is clear. If they are the same, we then compare the least significant bits (<code>a0<\/code> vs. <code>b0<\/code>).<\/li>\n<\/ul>\n\n\n\n<p>This logic leads to compact expressions using AND, OR, NOT, and XNOR gates. To minimize gates, shared terms such as <code>(a1 \u2261 b1)<\/code> can be computed once and reused.<\/p>\n\n\n\n<p>In Logisim, these expressions can be directly implemented with basic logic blocks. Testing the circuit against the truth table verifies correctness. This exercise is a fundamental example of digital logic design, showing how to translate a logical requirement into a functional circuit using Boolean algebra and minimization techniques.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-152.jpeg\" alt=\"\" class=\"wp-image-222290\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Design a 2-bit comparator. The comparator input has two 2-bit numbers A and B. A consists of 2 bits a1 and a0. B consists of 2 bits b1 and b0. The comparator has three outputs Z2, Z1, Z0. a) Show the truth table for this comparator b) Write the logic equation for Z2 that are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-222289","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222289","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=222289"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222289\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222289"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222289"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}