{"id":3202,"date":"2025-06-06T04:03:20","date_gmt":"2025-06-06T04:03:20","guid":{"rendered":"https:\/\/diznr.com\/?p=3202"},"modified":"2025-06-06T04:03:20","modified_gmt":"2025-06-06T04:03:20","slug":"boolean-algebra-introduction-closed-identity-commutative-distributive-complement-least-at","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/boolean-algebra-introduction-closed-identity-commutative-distributive-complement-least-at\/","title":{"rendered":"Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least."},"content":{"rendered":"<p>Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.<\/p>\n<p>[fvplayer id=&#8221;297&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"41\"><strong data-start=\"4\" data-end=\"39\">Introduction to Boolean Algebra<\/strong><\/h3>\n<p data-start=\"43\" data-end=\"267\"><strong data-start=\"43\" data-end=\"62\">Boolean Algebra<\/strong> \u090f\u0915 Mathematical System \u0939\u0948 \u091c\u093f\u0938\u0947 <strong data-start=\"94\" data-end=\"110\">George Boole<\/strong> \u0928\u0947 \u0935\u093f\u0915\u0938\u093f\u0924 \u0915\u093f\u092f\u093e \u0925\u093e\u0964 \u092f\u0939 <strong data-start=\"133\" data-end=\"159\">Binary System (0 \u0914\u0930 1)<\/strong> \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 <strong data-start=\"192\" data-end=\"247\">Digital Logic, Circuit Design, and Computer Science<\/strong> \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4 data-start=\"269\" data-end=\"315\"><strong data-start=\"274\" data-end=\"313\">Basic Operations in Boolean Algebra<\/strong><\/h4>\n<p data-start=\"316\" data-end=\"365\">Boolean Algebra \u092e\u0947\u0902 \u0924\u0940\u0928 \u092e\u0941\u0916\u094d\u092f \u0911\u092a\u0930\u0947\u0936\u0902\u0938 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<ol data-start=\"366\" data-end=\"555\">\n<li data-start=\"366\" data-end=\"428\"><strong data-start=\"369\" data-end=\"380\">AND (\u22c5)<\/strong> \u2192 Output 1 \u0924\u092d\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c \u0926\u094b\u0928\u094b\u0902 Inputs 1 \u0939\u094b\u0902\u0964<\/li>\n<li data-start=\"429\" data-end=\"486\"><strong data-start=\"432\" data-end=\"442\">OR (+)<\/strong> \u2192 Output 1 \u0939\u094b\u0924\u093e \u0939\u0948 \u0905\u0917\u0930 \u0915\u094b\u0908 \u092d\u0940 Input 1 \u0939\u094b\u0964<\/li>\n<li data-start=\"487\" data-end=\"555\"><strong data-start=\"490\" data-end=\"503\">NOT ( \u0305 )<\/strong> \u2192 Input \u0915\u093e \u0909\u0932\u094d\u091f\u093e Output \u0926\u0947\u0924\u093e \u0939\u0948 (0 \u2192 1 \u0914\u0930 1 \u2192 0)\u0964<\/li>\n<\/ol>\n<h3 data-start=\"562\" data-end=\"636\"><strong data-start=\"565\" data-end=\"634\">Boolean Algebra \u0915\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923 (Properties of Boolean Algebra)<\/strong><\/h3>\n<h3 data-start=\"638\" data-end=\"685\"><strong data-start=\"642\" data-end=\"683\">\u00a0Closure Property (\u0938\u092e\u094d\u092a\u0942\u0930\u094d\u0923\u0924\u093e \u0917\u0941\u0923)<\/strong><\/h3>\n<p data-start=\"686\" data-end=\"848\">Boolean Algebra \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0926\u094b Boolean \u0935\u0948\u0930\u093f\u090f\u092c\u0932\u094d\u0938 \u0915\u094b AND (+) \u092f\u093e OR (\u22c5) \u0915\u0930\u0928\u0947 \u092a\u0930 \u092d\u0940 Boolean \u0935\u0948\u0932\u094d\u092f\u0942 (0 \u092f\u093e 1) \u0939\u0940 \u092e\u093f\u0932\u0924\u0940 \u0939\u0948\u0964<br data-start=\"803\" data-end=\"806\" \/>\u00a0\u092f\u0926\u093f A \u0914\u0930 B Boolean Variables \u0939\u0948\u0902, \u0924\u094b:<\/p>\n<ul data-start=\"849\" data-end=\"957\">\n<li data-start=\"849\" data-end=\"902\"><strong data-start=\"851\" data-end=\"860\">A + B<\/strong> (OR Operation) = Boolean Value (0 \u092f\u093e 1)<\/li>\n<li data-start=\"903\" data-end=\"957\"><strong data-start=\"905\" data-end=\"914\">A \u22c5 B<\/strong> (AND Operation) = Boolean Value (0 \u092f\u093e 1)<\/li>\n<\/ul>\n<p data-start=\"959\" data-end=\"975\"><strong data-start=\"961\" data-end=\"973\">Example:<\/strong><\/p>\n<ul data-start=\"976\" data-end=\"1042\">\n<li data-start=\"976\" data-end=\"1008\"><strong data-start=\"978\" data-end=\"991\">1 + 0 = 1<\/strong> (OR operation)<\/li>\n<li data-start=\"1009\" data-end=\"1042\"><strong data-start=\"1011\" data-end=\"1024\">1 \u22c5 0 = 0<\/strong> (AND operation)<\/li>\n<\/ul>\n<p data-start=\"1044\" data-end=\"1128\">\u00a0Boolean Algebra <strong data-start=\"1062\" data-end=\"1072\">Closed<\/strong> \u0930\u0939\u0924\u093e \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u0939\u092e\u0947\u0936\u093e 0 \u0914\u0930 1 \u0915\u0947 \u0926\u093e\u092f\u0930\u0947 \u092e\u0947\u0902 \u0930\u0939\u0924\u093e \u0939\u0948\u0964<\/p>\n<h3 data-start=\"1135\" data-end=\"1180\"><strong data-start=\"1139\" data-end=\"1178\">\u00a0Identity Property (\u092a\u0930\u093f\u091a\u093e\u092f\u0915 \u0917\u0941\u0923)<\/strong><\/h3>\n<p data-start=\"1181\" data-end=\"1268\">Identity Property \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f <strong data-start=\"1211\" data-end=\"1221\">0 \u0914\u0930 1<\/strong>, OR \u0914\u0930 AND \u0911\u092a\u0930\u0947\u0936\u0928 \u092e\u0947\u0902 \u0915\u0948\u0938\u0947 \u0935\u094d\u092f\u0935\u0939\u093e\u0930 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"1270\" data-end=\"1301\"><strong data-start=\"1273\" data-end=\"1299\">For AND (\u22c5) Operation:<\/strong><\/p>\n<ul data-start=\"1302\" data-end=\"1337\">\n<li data-start=\"1302\" data-end=\"1319\"><strong data-start=\"1304\" data-end=\"1317\">A \u22c5 1 = A<\/strong><\/li>\n<li data-start=\"1320\" data-end=\"1337\"><strong data-start=\"1322\" data-end=\"1335\">A \u22c5 0 = 0<\/strong><\/li>\n<\/ul>\n<p data-start=\"1339\" data-end=\"1369\"><strong data-start=\"1342\" data-end=\"1367\">For OR (+) Operation:<\/strong><\/p>\n<ul data-start=\"1370\" data-end=\"1405\">\n<li data-start=\"1370\" data-end=\"1387\"><strong data-start=\"1372\" data-end=\"1385\">A + 0 = A<\/strong><\/li>\n<li data-start=\"1388\" data-end=\"1405\"><strong data-start=\"1390\" data-end=\"1403\">A + 1 = 1<\/strong><\/li>\n<\/ul>\n<p data-start=\"1407\" data-end=\"1423\"><strong data-start=\"1409\" data-end=\"1421\">Example:<\/strong><\/p>\n<ul data-start=\"1424\" data-end=\"1477\">\n<li data-start=\"1424\" data-end=\"1441\"><strong data-start=\"1426\" data-end=\"1439\">1 \u22c5 1 = 1<\/strong><\/li>\n<li data-start=\"1442\" data-end=\"1459\"><strong data-start=\"1444\" data-end=\"1457\">1 + 0 = 1<\/strong><\/li>\n<li data-start=\"1460\" data-end=\"1477\"><strong data-start=\"1462\" data-end=\"1475\">0 \u22c5 1 = 0<\/strong><\/li>\n<\/ul>\n<p data-start=\"1479\" data-end=\"1554\">\u00a01 \u0914\u0930 0 \u0915\u0940 \u092f\u0939 \u0935\u093f\u0936\u0947\u0937\u0924\u093e Boolean Algebra \u092e\u0947\u0902 \u092a\u0939\u091a\u093e\u0928 (Identity) \u092c\u0928\u093e\u090f \u0930\u0916\u0924\u0940 \u0939\u0948\u0964<\/p>\n<h3 data-start=\"1561\" data-end=\"1609\"><strong data-start=\"1565\" data-end=\"1607\">\u00a0Commutative Property (\u0905\u0935\u093f\u0928\u093f\u092f\u092e \u0917\u0941\u0923)<\/strong><\/h3>\n<p data-start=\"1610\" data-end=\"1742\">Commutative Property \u0915\u0939\u0924\u0940 \u0939\u0948 \u0915\u093f Boolean Algebra \u092e\u0947\u0902 OR \u0914\u0930 AND \u0915\u0947 \u0932\u093f\u090f <strong data-start=\"1679\" data-end=\"1739\">Operands \u0915\u0940 Order \u092c\u0926\u0932\u0928\u0947 \u0938\u0947 Result \u092a\u0930 \u0915\u094b\u0908 \u092b\u0930\u094d\u0915 \u0928\u0939\u0940\u0902 \u092a\u0921\u093c\u0924\u093e<\/strong>\u0964<\/p>\n<p data-start=\"1744\" data-end=\"1775\"><strong data-start=\"1747\" data-end=\"1773\">For AND (\u22c5) Operation:<\/strong><\/p>\n<ul data-start=\"1776\" data-end=\"1797\">\n<li data-start=\"1776\" data-end=\"1797\"><strong data-start=\"1778\" data-end=\"1795\">A \u22c5 B = B \u22c5 A<\/strong><\/li>\n<\/ul>\n<p data-start=\"1799\" data-end=\"1829\"><strong data-start=\"1802\" data-end=\"1827\">For OR (+) Operation:<\/strong><\/p>\n<ul data-start=\"1830\" data-end=\"1851\">\n<li data-start=\"1830\" data-end=\"1851\"><strong data-start=\"1832\" data-end=\"1849\">A + B = B + A<\/strong><\/li>\n<\/ul>\n<p data-start=\"1853\" data-end=\"1869\"><strong data-start=\"1855\" data-end=\"1867\">Example:<\/strong><\/p>\n<ul data-start=\"1870\" data-end=\"1921\">\n<li data-start=\"1870\" data-end=\"1895\"><strong data-start=\"1872\" data-end=\"1893\">1 + 0 = 0 + 1 = 1<\/strong><\/li>\n<li data-start=\"1896\" data-end=\"1921\"><strong data-start=\"1898\" data-end=\"1919\">1 \u22c5 0 = 0 \u22c5 1 = 0<\/strong><\/li>\n<\/ul>\n<p data-start=\"1923\" data-end=\"1989\">\u00a0Commutative Property Boolean Operations \u0915\u094b \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930\u0924\u093e \u0926\u0947\u0924\u0940 \u0939\u0948\u0964<\/p>\n<h3 data-start=\"1996\" data-end=\"2043\"><strong data-start=\"2000\" data-end=\"2041\">\u00a0Distributive Property (\u0935\u093f\u0924\u0930\u0923 \u0917\u0941\u0923)<\/strong><\/h3>\n<p data-start=\"2044\" data-end=\"2155\">Distributive Property \u092f\u0939 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f \u0915\u093f\u0938\u0940 Boolean Expression \u092e\u0947\u0902 OR \u0914\u0930 AND \u0915\u0948\u0938\u0947 \u0935\u093f\u0924\u0930\u093f\u0924 (distribute) \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"2157\" data-end=\"2178\"><strong data-start=\"2160\" data-end=\"2176\">AND over OR:<\/strong><\/p>\n<ul data-start=\"2179\" data-end=\"2218\">\n<li data-start=\"2179\" data-end=\"2218\"><strong data-start=\"2181\" data-end=\"2216\">A \u22c5 (B + C) = (A \u22c5 B) + (A \u22c5 C)<\/strong><\/li>\n<\/ul>\n<p data-start=\"2220\" data-end=\"2241\"><strong data-start=\"2223\" data-end=\"2239\">OR over AND:<\/strong><\/p>\n<ul data-start=\"2242\" data-end=\"2281\">\n<li data-start=\"2242\" data-end=\"2281\"><strong data-start=\"2244\" data-end=\"2279\">A + (B \u22c5 C) = (A + B) \u22c5 (A + C)<\/strong><\/li>\n<\/ul>\n<p data-start=\"2283\" data-end=\"2299\"><strong data-start=\"2285\" data-end=\"2297\">Example:<\/strong><\/p>\n<ul data-start=\"2300\" data-end=\"2371\">\n<li data-start=\"2300\" data-end=\"2339\"><strong data-start=\"2302\" data-end=\"2337\">1 \u22c5 (0 + 1) = (1 \u22c5 0) + (1 \u22c5 1)<\/strong><\/li>\n<li data-start=\"2340\" data-end=\"2371\"><strong data-start=\"2342\" data-end=\"2357\">= 0 + 1 = 1<\/strong> (LHS = RHS)<\/li>\n<\/ul>\n<p data-start=\"2373\" data-end=\"2470\">\u00a0Distributive Property Boolean Expressions \u0915\u094b Factorization \u0914\u0930 Expansion \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964<\/p>\n<h3 data-start=\"2477\" data-end=\"2524\"><strong data-start=\"2481\" data-end=\"2522\">\u00a0Complement Property (\u092a\u0930\u093f\u092a\u0942\u0930\u0915 \u0917\u0941\u0923)<\/strong><\/h3>\n<p data-start=\"2525\" data-end=\"2630\">Complement Property \u092f\u0939 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f \u0915\u093f\u0938\u0940 Boolean Variable \u0915\u093e \u0909\u0932\u094d\u091f\u093e \u0915\u0930\u0928\u0947 (NOT Operation) \u0938\u0947 \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<p data-start=\"2632\" data-end=\"2647\"><strong data-start=\"2635\" data-end=\"2645\">Rules:<\/strong><\/p>\n<ol data-start=\"2648\" data-end=\"2751\">\n<li data-start=\"2648\" data-end=\"2667\"><strong data-start=\"2651\" data-end=\"2665\">A + A\u0305 = 1<\/strong><\/li>\n<li data-start=\"2668\" data-end=\"2687\"><strong data-start=\"2671\" data-end=\"2685\">A \u22c5 A\u0305 = 0<\/strong><\/li>\n<li data-start=\"2688\" data-end=\"2751\"><strong data-start=\"2691\" data-end=\"2704\">(A\u0305)\u0305 = A<\/strong> (Double Complement gives the original value)<\/li>\n<\/ol>\n<p data-start=\"2753\" data-end=\"2769\"><strong data-start=\"2755\" data-end=\"2767\">Example:<\/strong><\/p>\n<ul data-start=\"2770\" data-end=\"2856\">\n<li data-start=\"2770\" data-end=\"2802\">\u0905\u0917\u0930 <strong data-start=\"2776\" data-end=\"2785\">A = 1<\/strong>, \u0924\u094b <strong data-start=\"2790\" data-end=\"2800\">A\u0305 = 0<\/strong><\/li>\n<li data-start=\"2803\" data-end=\"2829\"><strong data-start=\"2805\" data-end=\"2827\">A + A\u0305 = 1 + 0 = 1<\/strong><\/li>\n<li data-start=\"2830\" data-end=\"2856\"><strong data-start=\"2832\" data-end=\"2854\">A \u22c5 A\u0305 = 1 \u22c5 0 = 0<\/strong><\/li>\n<\/ul>\n<p data-start=\"2858\" data-end=\"2933\">\u00a0Complement Property Boolean Functions \u0915\u094b Simplify \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964<\/p>\n<h3 data-start=\"2940\" data-end=\"2965\"><strong data-start=\"2943\" data-end=\"2963\">Summary (\u0938\u093e\u0930\u093e\u0902\u0936)<\/strong><\/h3>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"2967\" data-end=\"3414\">\n<thead data-start=\"2967\" data-end=\"3028\">\n<tr data-start=\"2967\" data-end=\"3028\">\n<th data-start=\"2967\" data-end=\"2987\"><strong data-start=\"2969\" data-end=\"2981\">Property<\/strong><\/th>\n<th data-start=\"2987\" data-end=\"3013\"><strong data-start=\"2989\" data-end=\"3003\">Expression<\/strong><\/th>\n<th data-start=\"3013\" data-end=\"3028\"><strong data-start=\"3015\" data-end=\"3026\">Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"3086\" data-end=\"3414\">\n<tr data-start=\"3086\" data-end=\"3142\">\n<td><strong data-start=\"3088\" data-end=\"3099\">Closure<\/strong><\/td>\n<td>A + B, A \u22c5 B \u2208 {0,1}<\/td>\n<td>1 + 0 = 1<\/td>\n<\/tr>\n<tr data-start=\"3143\" data-end=\"3199\">\n<td><strong data-start=\"3145\" data-end=\"3157\">Identity<\/strong><\/td>\n<td>A + 0 = A, A \u22c5 1 = A<\/td>\n<td>1 \u22c5 1 = 1<\/td>\n<\/tr>\n<tr data-start=\"3200\" data-end=\"3267\">\n<td><strong data-start=\"3202\" data-end=\"3217\">Commutative<\/strong><\/td>\n<td>A + B = B + A, A \u22c5 B = B \u22c5 A<\/td>\n<td>0 + 1 = 1 + 0<\/td>\n<\/tr>\n<tr data-start=\"3268\" data-end=\"3356\">\n<td><strong data-start=\"3270\" data-end=\"3286\">Distributive<\/strong><\/td>\n<td>A \u22c5 (B + C) = (A \u22c5 B) + (A \u22c5 C)<\/td>\n<td>1 \u22c5 (0 + 1) = (1 \u22c5 0) + (1 \u22c5 1)<\/td>\n<\/tr>\n<tr data-start=\"3357\" data-end=\"3414\">\n<td><strong data-start=\"3359\" data-end=\"3373\">Complement<\/strong><\/td>\n<td>A + A\u0305 = 1, A \u22c5 A\u0305 = 0<\/td>\n<td>1 + 0 = 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p data-start=\"3416\" data-end=\"3536\"><strong data-start=\"3418\" data-end=\"3534\">Boolean Algebra \u0915\u093e \u0909\u092a\u092f\u094b\u0917 Digital Circuits, Logic Gates, and Computer Algorithms \u0915\u094b Design \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n<p data-start=\"3538\" data-end=\"3627\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"3541\" data-end=\"3627\" data-is-last-node=\"\">\u0915\u094d\u092f\u093e \u0906\u092a Boolean Algebra \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u0915\u093f\u0938\u0940 \u0914\u0930 Concept \u0915\u094b \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u0938\u092e\u091d\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902?<\/strong><\/p>\n<h3 data-start=\"3538\" data-end=\"3627\"><a href=\"https:\/\/www.pvpsiddhartha.ac.in\/dep_it\/lecture%20notes\/DSD\/unit2.pdf\" target=\"_blank\" rel=\"noopener\">Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.uobabylon.edu.iq\/eprints\/publication_2_28322_1447.pdf\" target=\"_blank\" rel=\"noopener\">BOOLEAN ALGEBRA 2.1 Introduction<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"http:\/\/www.ilovemaths.in\/wp-content\/uploads\/maths-content\/classes11th12th\/isc12-boolean-algebra.pdf\" target=\"_blank\" rel=\"noopener\">3 Boolean Algebra &#8211; I Love Maths<\/a><\/h3>\n<p data-start=\"0\" data-end=\"225\">Here&#8217;s a concise <strong data-start=\"17\" data-end=\"52\">introduction to Boolean Algebra<\/strong> with key properties like <strong data-start=\"78\" data-end=\"126\">Closure, Identity, Commutative, Distributive<\/strong>, and <strong data-start=\"132\" data-end=\"146\">Complement<\/strong> \u2014 essential for understanding digital logic and computer science fundamentals.<\/p>\n<hr data-start=\"227\" data-end=\"230\" \/>\n<h2 data-start=\"232\" data-end=\"272\">\ud83e\uddee <strong data-start=\"238\" data-end=\"272\">Boolean Algebra \u2013 Introduction<\/strong><\/h2>\n<p data-start=\"274\" data-end=\"321\">Boolean algebra deals with <strong data-start=\"301\" data-end=\"318\">binary values<\/strong>:<\/p>\n<ul data-start=\"322\" data-end=\"433\">\n<li data-start=\"322\" data-end=\"339\">\n<p data-start=\"324\" data-end=\"339\"><strong data-start=\"324\" data-end=\"329\">0<\/strong> (False)<\/p>\n<\/li>\n<li data-start=\"340\" data-end=\"388\">\n<p data-start=\"342\" data-end=\"388\"><strong data-start=\"342\" data-end=\"347\">1<\/strong> (True)<br data-start=\"354\" data-end=\"357\" \/>Using <strong data-start=\"363\" data-end=\"385\">logical operations<\/strong>:<\/p>\n<\/li>\n<li data-start=\"389\" data-end=\"404\">\n<p data-start=\"391\" data-end=\"404\"><strong data-start=\"391\" data-end=\"402\">AND (\u00b7)<\/strong><\/p>\n<\/li>\n<li data-start=\"405\" data-end=\"419\">\n<p data-start=\"407\" data-end=\"419\"><strong data-start=\"407\" data-end=\"417\">OR (+)<\/strong><\/p>\n<\/li>\n<li data-start=\"420\" data-end=\"433\">\n<p data-start=\"422\" data-end=\"433\"><strong data-start=\"422\" data-end=\"433\">NOT (&#8216;)<\/strong><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"435\" data-end=\"438\" \/>\n<h2 data-start=\"440\" data-end=\"482\">\u2705 <strong data-start=\"445\" data-end=\"482\">Key Properties in Boolean Algebra<\/strong><\/h2>\n<h3 data-start=\"484\" data-end=\"512\">1\ufe0f\u20e3 <strong data-start=\"492\" data-end=\"512\">Closure Property<\/strong><\/h3>\n<ul data-start=\"513\" data-end=\"644\">\n<li data-start=\"513\" data-end=\"570\">\n<p data-start=\"515\" data-end=\"570\">Boolean algebra is <strong data-start=\"534\" data-end=\"544\">closed<\/strong> under AND, OR, and NOT.<\/p>\n<\/li>\n<li data-start=\"571\" data-end=\"644\">\n<p data-start=\"573\" data-end=\"644\">Example: If <strong data-start=\"585\" data-end=\"601\">A, B \u2208 {0,1}<\/strong>, then <strong data-start=\"608\" data-end=\"625\">A + B \u2208 {0,1}<\/strong>, <strong data-start=\"627\" data-end=\"644\">A \u00b7 B \u2208 {0,1}<\/strong><\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"646\" data-end=\"675\">2\ufe0f\u20e3 <strong data-start=\"654\" data-end=\"675\">Identity Property<\/strong><\/h3>\n<ul data-start=\"676\" data-end=\"729\">\n<li data-start=\"676\" data-end=\"703\">\n<p data-start=\"678\" data-end=\"703\">For OR:\u2003\u2003\u2003<strong data-start=\"688\" data-end=\"701\">A + 0 = A<\/strong><\/p>\n<\/li>\n<li data-start=\"704\" data-end=\"729\">\n<p data-start=\"706\" data-end=\"729\">For AND:\u2003\u2003<strong data-start=\"716\" data-end=\"729\">A \u00b7 1 = A<\/strong><\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"731\" data-end=\"763\">3\ufe0f\u20e3 <strong data-start=\"739\" data-end=\"763\">Commutative Property<\/strong><\/h3>\n<ul data-start=\"764\" data-end=\"842\">\n<li data-start=\"764\" data-end=\"785\">\n<p data-start=\"766\" data-end=\"785\"><strong data-start=\"766\" data-end=\"783\">A + B = B + A<\/strong><\/p>\n<\/li>\n<li data-start=\"786\" data-end=\"842\">\n<p data-start=\"788\" data-end=\"842\"><strong data-start=\"788\" data-end=\"805\">A \u00b7 B = B \u00b7 A<\/strong><br data-start=\"805\" data-end=\"808\" \/>Order of operation doesn\u2019t matter.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"844\" data-end=\"877\">4\ufe0f\u20e3 <strong data-start=\"852\" data-end=\"877\">Distributive Property<\/strong><\/h3>\n<ul data-start=\"878\" data-end=\"997\">\n<li data-start=\"878\" data-end=\"934\">\n<p data-start=\"880\" data-end=\"934\">AND distributes over OR:\u2003<strong data-start=\"905\" data-end=\"932\">A \u00b7 (B + C) = A\u00b7B + A\u00b7C<\/strong><\/p>\n<\/li>\n<li data-start=\"935\" data-end=\"997\">\n<p data-start=\"937\" data-end=\"997\">OR distributes over AND:\u2003<strong data-start=\"962\" data-end=\"997\">A + (B \u00b7 C) = (A + B) \u00b7 (A + C)<\/strong><\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"999\" data-end=\"1025\">5\ufe0f\u20e3 <strong data-start=\"1007\" data-end=\"1025\">Complement Law<\/strong><\/h3>\n<ul data-start=\"1026\" data-end=\"1102\">\n<li data-start=\"1026\" data-end=\"1044\">\n<p data-start=\"1028\" data-end=\"1044\"><strong data-start=\"1028\" data-end=\"1042\">A + A&#8217; = 1<\/strong><\/p>\n<\/li>\n<li data-start=\"1045\" data-end=\"1102\">\n<p data-start=\"1047\" data-end=\"1102\"><strong data-start=\"1047\" data-end=\"1061\">A \u00b7 A&#8217; = 0<\/strong><br data-start=\"1061\" data-end=\"1064\" \/>Where A&#8217; is the <strong data-start=\"1080\" data-end=\"1094\">complement<\/strong> (NOT A)<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1104\" data-end=\"1107\" \/>\n<h3 data-start=\"1109\" data-end=\"1140\">\ud83e\udde0 Bonus: Other Useful Laws<\/h3>\n<ul data-start=\"1141\" data-end=\"1255\">\n<li data-start=\"1141\" data-end=\"1181\">\n<p data-start=\"1143\" data-end=\"1181\"><strong data-start=\"1143\" data-end=\"1157\">Idempotent<\/strong>:\u2003A + A = A,\u2003A \u00b7 A = A<\/p>\n<\/li>\n<li data-start=\"1182\" data-end=\"1222\">\n<p data-start=\"1184\" data-end=\"1222\"><strong data-start=\"1184\" data-end=\"1198\">Domination<\/strong>:\u2003A + 1 = 1,\u2003A \u00b7 0 = 0<\/p>\n<\/li>\n<li data-start=\"1223\" data-end=\"1255\">\n<p data-start=\"1225\" data-end=\"1255\"><strong data-start=\"1225\" data-end=\"1244\">Double Negation<\/strong>:\u2003(A&#8217;)&#8217; = A<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1257\" data-end=\"1260\" \/>\n<p data-start=\"1262\" data-end=\"1355\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like <span class=\"decoration-token-text-secondary hover:text-token-text-secondary cursor-pointer underline decoration-dotted decoration-[12%] underline-offset-4 transition-colors duration-200 ease-in-out\">truth table examples<\/span>, <span class=\"decoration-token-text-secondary hover:text-token-text-secondary cursor-pointer underline decoration-dotted decoration-[12%] underline-offset-4 transition-colors duration-200 ease-in-out\">Boolean laws chart<\/span>, or <span class=\"decoration-token-text-secondary hover:text-token-text-secondary cursor-pointer underline decoration-dotted decoration-[12%] underline-offset-4 transition-colors duration-200 ease-in-out\">practice problems<\/span>?<\/p>\n<h3 data-start=\"1262\" data-end=\"1355\"><a href=\"https:\/\/www.pvpsiddhartha.ac.in\/dep_it\/lecture%20notes\/FDLD_21\/UNIT-2.pdf\" target=\"_blank\" rel=\"noopener\">Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.iitmanagement.com\/images\/Gallery\/DIP-EE-4TH%20SEM%20-%20DE.pdf\" target=\"_blank\" rel=\"noopener\">UNIT &#8211; 1 NUMBER SYSTEMS &amp; BOOLEAN ALGEBRA<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"http:\/\/www.cectl.ac.in\/images\/pdf_docs\/studymaterial\/cse\/s3\/dcs4.pdf\" target=\"_blank\" rel=\"noopener\">LATTICE AND BOOLEAN ALGEBRA<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least. [fvplayer id=&#8221;297&#8243;] Introduction to Boolean Algebra Boolean Algebra \u090f\u0915 Mathematical System \u0939\u0948 \u091c\u093f\u0938\u0947 George Boole \u0928\u0947 \u0935\u093f\u0915\u0938\u093f\u0924 \u0915\u093f\u092f\u093e \u0925\u093e\u0964 \u092f\u0939 Binary System (0 \u0914\u0930 1) \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 Digital Logic, Circuit Design, and Computer Science \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 Basic Operations [&hellip;]<\/p>\n","protected":false},"author":66,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111],"tags":[],"class_list":["post-3202","post","type-post","status-publish","format-standard","hentry","category-digital-electronics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=3202"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3202\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3202"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3202"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}