{"id":239133,"date":"2025-07-02T18:23:21","date_gmt":"2025-07-02T18:23:21","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=239133"},"modified":"2025-07-02T18:23:23","modified_gmt":"2025-07-02T18:23:23","slug":"the-truth-table-for-p-v-q-v-p-a%cb%86%c2%a7-r-is-the-same-as-the-truth-table-for","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/02\/the-truth-table-for-p-v-q-v-p-a%cb%86%c2%a7-r-is-the-same-as-the-truth-table-for\/","title":{"rendered":"The truth table for (p V q) v (p \u00e2\u02c6\u00a7 r) is the same as the truth table for"},"content":{"rendered":"\n

question The truth table for (p V q) v (p \u00e2\u02c6\u00a7 r) is the same as the truth table for: p v q \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 q) \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 r) \u00e2\u02c6\u00a7 (p v r)
Time left 1.23 * 12 Question 9 Not yet answered Marked out of 3 00 Flag question The truth table for (p V q) v (p \u00e2\u02c6\u00a7 r) is the same as the truth table for: p v q \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 q) \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 r) \u00e2\u02c6\u00a7 (p v r)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The correct answer is a. p \u2228 q<\/strong>.<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

To determine which expression has the same truth table as (p \u2228 q) \u2228 (p \u2227 r), we need to find a logically equivalent expression. This can be done by either using the laws of propositional logic to simplify the expression or by constructing and comparing truth tables.<\/p>\n\n\n\n

Method 1: Using Laws of Propositional Logic<\/strong><\/p>\n\n\n\n

This is often the most direct method. The goal is to simplify the original expression: (p \u2228 q) \u2228 (p \u2227 r).<\/p>\n\n\n\n

    \n
  1. Original Expression:<\/strong>\u00a0(p \u2228 q) \u2228 (p \u2227 r)<\/li>\n\n\n\n
  2. Apply the Associative and Commutative Laws:<\/strong>\u00a0The\u00a0\u2228\u00a0(OR) operator is associative and commutative, which means we can reorder and regroup the terms. Let’s group the terms involving\u00a0p\u00a0together.
    (p \u2228 q) \u2228 (p \u2227 r) \u2261 p \u2228 q \u2228 (p \u2227 r) \u2261 q \u2228 (p \u2228 (p \u2227 r))<\/li>\n\n\n\n
  3. Apply the Absorption Law:<\/strong>\u00a0The Absorption Law states that\u00a0A \u2228 (A \u2227 B) \u2261 A. In the sub-expression\u00a0p \u2228 (p \u2227 r), we can see this pattern where\u00a0A = p\u00a0and\u00a0B = r. Therefore, the expression\u00a0p \u2228 (p \u2227 r)\u00a0simplifies to just\u00a0p.<\/li>\n\n\n\n
  4. Substitute and Final Simplification:<\/strong>\u00a0Now we substitute this simplified part back into the expression from step 2:
    q \u2228 (p)<\/li>\n\n\n\n
  5. Using the Commutative Law,\u00a0q \u2228 p\u00a0is equivalent to\u00a0p \u2228 q.<\/li>\n<\/ol>\n\n\n\n

    Thus, the original complex expression simplifies to p \u2228 q, which matches option a<\/strong>.<\/p>\n\n\n\n

    Method 2: Using a Truth Table<\/strong><\/p>\n\n\n\n

    We can verify this result by constructing a truth table for the original expression and for the simplified expression p \u2228 q. If their final truth values are identical for all possible inputs, they are equivalent.<\/p>\n\n\n\n

    Let’s create the truth table for (p \u2228 q) \u2228 (p \u2227 r):<\/p>\n\n\n\n

    p<\/td>q<\/td>r<\/td>p \u2228 q<\/td>p \u2227 r<\/td>(p \u2228 q) \u2228 (p \u2227 r)<\/strong><\/td>p \u2228 q<\/strong><\/td><\/tr>
    T<\/td>T<\/td>T<\/td>T<\/td>T<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    T<\/td>T<\/td>F<\/td>T<\/td>F<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    T<\/td>F<\/td>T<\/td>T<\/td>T<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    T<\/td>F<\/td>F<\/td>T<\/td>F<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    F<\/td>T<\/td>T<\/td>T<\/td>F<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    F<\/td>T<\/td>F<\/td>T<\/td>F<\/td>T<\/strong><\/td>T<\/strong><\/td><\/tr>
    F<\/td>F<\/td>T<\/td>F<\/td>F<\/td>F<\/strong><\/td>F<\/strong><\/td><\/tr>
    F<\/td>F<\/td>F<\/td>F<\/td>F<\/td>F<\/strong><\/td>F<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    As shown in the table, the final column for the original expression (p \u2228 q) \u2228 (p \u2227 r) is identical to the column for p \u2228 q. This confirms that they are logically equivalent. Therefore, they have the same truth table.thumb_upthumb_down<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    question The truth table for (p V q) v (p \u00e2\u02c6\u00a7 r) is the same as the truth table for: p v q \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 q) \u00e2\u02c6\u00a7 (p \u00e2\u02c6\u00a7 r) \u00e2\u02c6\u00a7 (p v r)Time left 1.23 * 12 Question 9 Not yet answered Marked out of 3 00 Flag question The truth table for […]<\/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-239133","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239133","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=239133"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/239133\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=239133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=239133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=239133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}