{"id":7810,"date":"2022-04-25T13:09:46","date_gmt":"2022-04-25T07:39:46","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7810"},"modified":"2026-03-03T05:26:05","modified_gmt":"2026-03-02T23:56:05","slug":"what-is-boolean-algebra","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-boolean-algebra\/","title":{"rendered":"What is Boolean Algebra?"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li>Boolean algebra which deals&nbsp; with variables and binary numbers.<\/li>\n\n\n\n<li>This algebra is named after mathematician George Boole who developed this in 1854.<\/li>\n\n\n\n<li>The variables considered for simplification are called Boolean Variables.<\/li>\n\n\n\n<li>The value of voltage when HIGH is considered 1 and when LOW is considered as 0.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Basic Laws of Boolean Algebra<\/h2>\n\n\n\n<p>Here, the Boolean postulates and basic laws that are used are given underneath.<\/p>\n\n\n\n<p><strong>Boolean Postulates<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Considering the binary numbers 0 and 1, Boolean variable (x) and its complement (x\u2019).<\/li>\n\n\n\n<li>They are known as <strong>literal<\/strong>.<\/li>\n\n\n\n<li>The possible <strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Logic_gate\">logical OR<\/a><\/strong> operations are:<\/li>\n<\/ul>\n\n\n\n<p>x + 0 = x<\/p>\n\n\n\n<p>Also, x + 1 = 1<\/p>\n\n\n\n<p>x + x = x<\/p>\n\n\n\n<p>And x + x\u2019 = 1<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Similarly, the possible <strong>logical AND<\/strong> operations are:<\/li>\n<\/ul>\n\n\n\n<p>x.1 = x<\/p>\n\n\n\n<p>x.0 = 0<\/p>\n\n\n\n<p>x.x = x<\/p>\n\n\n\n<p>x.x\u2019 = 0<\/p>\n\n\n\n<p>We can verify these postulates by simply replacing the Boolean variables with 0 or 1.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Basic Laws<\/h2>\n\n\n\n<p>The basic laws of Boolean Algebra are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Commutative law<\/li>\n\n\n\n<li>Associative law<\/li>\n\n\n\n<li>Distributive law<\/li>\n<\/ul>\n\n\n\n<p><strong>Commutative Law<\/strong><\/p>\n\n\n\n<p>According to this law changing the sequence of the variables does not affect the output.&nbsp;&nbsp;<\/p>\n\n\n\n<p>The logical OR &amp; logical AND operations between x &amp; y are shown below<\/p>\n\n\n\n<p>x + y = y + x<\/p>\n\n\n\n<p>x.y = y.x<\/p>\n\n\n\n<p>The OR operation is represented by \u2018+\u2019 and AND operation by \u2018.\u2019&nbsp;<\/p>\n\n\n\n<p>This law exists for logical OR and AND operation.<\/p>\n\n\n\n<p><strong>Associative Law<\/strong><\/p>\n\n\n\n<p>This law states that the order of operation does not matter as the result remains unaffected. The equations below make it more clear.&nbsp;<\/p>\n\n\n\n<p>x + (y + z) = (x + y) + z<\/p>\n\n\n\n<p>x.(y.z) = (x.y).z<\/p>\n\n\n\n<p>This law exists for logical OR and AND operation.&nbsp;<\/p>\n\n\n\n<p><strong>Distributive Law<\/strong><\/p>\n\n\n\n<p>Basically this law helps to remove the brackets and solve the logical expression. The brackets can have either AND or OR logic. The distribution of logical OR &amp; logical AND operations between variables x, y &amp; z are :<\/p>\n\n\n\n<p>x.(y + z) = x.y + x.z<\/p>\n\n\n\n<p>x + (y.z) = (x + y).(x + z)<\/p>\n\n\n\n<p>It works good for logical OR and logical AND operations.&nbsp;<\/p>\n\n\n\n<p>The above laws are basic laws and can be verified by substituting the Boolean variable with \u20180\u2019 or \u20181\u2019.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Theorems of Boolean Algebra<\/h2>\n\n\n\n<p>Basic theorems of Boolean Algebra are<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Duality theorem<\/li>\n\n\n\n<li>De Morgan\u2019s theorem<\/li>\n<\/ul>\n\n\n\n<p><strong>Duality Theorem<\/strong><\/p>\n\n\n\n<p>As the name says we find the dual of the given Boolean expression. We replace AND operation with OR and vice versa. We replace all zeros by 1. There is always a <a href=\"https:\/\/smi-shipyard.com\/\">smi-shipard<\/a> dual function for every Boolean function.<\/p>\n\n\n\n<p>.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/D_xuHp16lk-IAM5dz4lbK9dZSMozp5eD3QcMD1OrFwGgAFvNVDCUKN_Ydw3GkPRNHWv7T_YZ3mPOfiXYn30bg9xdt1XgJsIRcYGagqcQ2DMCIwJlmuYHEK1lj9WVhTeuc4Q__g7I\" alt=\"\"\/><\/figure>\n\n\n\n<p>As seen from the above table there are two Boolean equations and are dual to one other.<\/p>\n\n\n\n<p>With the help of the duality theorem we can easily verify the Boolean equations of each group.<\/p>\n\n\n\n<p><strong>De Morgan\u2019s Theorem<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>De Morgan\u2019s Theorem helps us to find the complement of Boolean functions.<\/li>\n\n\n\n<li>According to this law the complement of the sum of variable is equal to product of complement of each variable.<\/li>\n\n\n\n<li>We can understand this law using two variables x and y as<\/li>\n<\/ul>\n\n\n\n<p>(x + y)\u2019 = x\u2019.y\u2019<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The second way or the dual of the above law with variables x and y is <\/li>\n<\/ul>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x.y)\u2019 = x\u2019 + y\u2019<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The complement of the product of two variables x and y is equal to the sum of complement of each variable.<\/li>\n\n\n\n<li>This theorem is applicable for more than two variables as well.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Boolean algebra which deals\u00a0 with variables and binary numbers. This algebra helps to simplify the Boolean variables<\/p>\n","protected":false},"author":3,"featured_media":7812,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[37],"tags":[],"class_list":["post-7810","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electronics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is Boolean Algebra? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"Boolean algebra which deals\u00a0 with variables and binary numbers. 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