{"id":12909,"date":"2023-01-02T15:11:30","date_gmt":"2023-01-02T12:11:30","guid":{"rendered":"https:\/\/starlanguageblog.com\/?p=12909"},"modified":"2023-01-02T15:11:30","modified_gmt":"2023-01-02T12:11:30","slug":"why-is-aab-ab","status":"publish","type":"post","link":"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/","title":{"rendered":"Why is \u0100+AB = \u0100+B ?"},"content":{"rendered":"

Why is \u0100+AB = \u0100+B ?<\/h1>\n

\u0100+AB = \u0100+B because the terms AB and B are equal. This follows from the fact that A+A = A, which is one of the axioms of Boolean algebra.<\/p>\n

In Boolean algebra, A+A = A is known as the “absorption” or “idempotent” law. It states that the OR of a value with itself is equal to the value itself. Similarly, A*A = A is known as the “idempotent” law of AND.<\/p>\n

Here’s a proof that AB = B using the axioms of Boolean algebra:<\/p>\n

    \n
  1. A+A = A (absorption law)<\/li>\n
  2. AB+A = A (substitution, using step 1)<\/li>\n
  3. AB+B = B (absorption law)<\/li>\n
  4. AB = B (cancellation law)<\/li>\n<\/ol>\n

    Here are some examples to illustrate the absorption law in Boolean algebra:<\/p>\n

    A+A = A<\/p>\n

    For example, if A represents the statement “It is raining outside,” then A+A would represent the statement “It is raining outside, OR it is raining outside.” This is clearly true, so A+A = A.<\/p>\n

    AB+A = A<\/p>\n

    For example, if A represents the statement “It is raining outside,” and B represents the statement “I have an umbrella,” then AB+A would represent the statement “It is raining outside AND I have an umbrella, OR it is raining outside.” This is also true, because if it is raining outside and I have an umbrella, then it is necessarily true<\/a> that it is raining outside. Therefore, AB+A = A.<\/p>\n

    AB+B = B<\/p>\n

    For example, if A represents the statement “It is raining outside,” and B represents the statement “I have an umbrella,” then AB+B would represent the statement “It is raining outside AND I have an umbrella, OR I have an umbrella.” This is true because if it is raining outside and I have an umbrella, then I have an umbrella (which is one of the terms being ORed). Similarly, if it is not raining outside, then having an umbrella is still true. Therefore, AB+B = B.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Why is \u0100+AB = \u0100+B ? \u0100+AB = \u0100+B because the terms AB and B are equal. This follows from the fact that A+A = A, which is one of the axioms of Boolean algebra. In Boolean algebra, A+A = A is known as the “absorption” or “idempotent” law. It states that the OR of […]<\/p>\n","protected":false},"author":1,"featured_media":12910,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2078],"tags":[2084],"yoast_head":"\nWhy is \u0100+AB = \u0100+B ? - Star Language Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Why is \u0100+AB = \u0100+B ? - Star Language Blog\" \/>\n<meta property=\"og:description\" content=\"Why is \u0100+AB = \u0100+B ? \u0100+AB = \u0100+B because the terms AB and B are equal. This follows from the fact that A+A = A, which is one of the axioms of Boolean algebra. In Boolean algebra, A+A = A is known as the “absorption” or “idempotent” law. It states that the OR of […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\" \/>\n<meta property=\"og:site_name\" content=\"Star Language Blog\" \/>\n<meta property=\"article:published_time\" content=\"2023-01-02T12:11:30+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2023\/01\/mathematics-g8b22faee6_640.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"640\" \/>\n\t<meta property=\"og:image:height\" content=\"448\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"author\" content=\"Starla\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Starla\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\"},\"author\":{\"name\":\"Starla\",\"@id\":\"https:\/\/www.starlanguageblog.com\/#\/schema\/person\/18c6c447348b68f36676245bfe3f83fc\"},\"headline\":\"Why is \u0100+AB = \u0100+B ?\",\"datePublished\":\"2023-01-02T12:11:30+00:00\",\"dateModified\":\"2023-01-02T12:11:30+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\"},\"wordCount\":323,\"publisher\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2023\/01\/mathematics-g8b22faee6_640.webp\",\"keywords\":[\"Why is \u0100+AB = \u0100+B ?\"],\"articleSection\":[\"Maths\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\",\"url\":\"https:\/\/www.starlanguageblog.com\/why-is-aab-ab\/\",\"name\":\"Why is \u0100+AB = \u0100+B ? 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