{"id":6879,"date":"2022-01-02T04:13:04","date_gmt":"2022-01-02T04:13:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6879"},"modified":"2022-09-09T00:29:50","modified_gmt":"2022-09-09T00:29:50","slug":"1-9-order-of-logical-operations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-9-order-of-logical-operations\/","title":{"raw":"1.9 Order of Logical Operations","rendered":"1.9 Order of Logical Operations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Commas in Logical Statements<\/li>\r\n \t<li>Order of Logical Operations\r\n<ul>\r\n \t<li>Create truth tables for complex statements using all operations (negations, conjunctions, disjunctions, conditionals, biconditionals)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen combining operations in logic, commas are important.\r\n<div class=\"textbox\">\r\n\r\n<strong>Commas in Logical Statements<\/strong>\r\n\r\nWhen we translate English statements into logical symbols, <strong>commas are used to show where the parentheses are placed<\/strong>.\r\n\r\n&nbsp;\r\n\r\nAn easy way to visualize it is to think in terms of the symbols.\r\n\r\n<strong>Example 1:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:\r\n<p style=\"text-align: center;\">P and Q, or R.<\/p>\r\nThe comma comes after the Q, so \u201cP and Q\u201d is inside parentheses.\r\n<p style=\"text-align: center;\">(P and Q), or R.<\/p>\r\nOr, if we write using all symbols, P and Q, or R becomes:\r\n<p style=\"text-align: center;\">[latex]\\left(P\\land Q\\right)\\lor{R}[\/latex].<\/p>\r\n&nbsp;\r\n\r\n<strong>Example 2:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:\r\n<p style=\"text-align: center;\">\u00a0If P then Q, if and only if R.<\/p>\r\nThe comma comes after the Q, so \u201cif P then Q\u201d is inside parentheses. Using symbols that looks like:\r\n<p style=\"text-align: center;\">[latex]\\left(P\\rightarrow Q\\right)\\leftrightarrow R[\/latex].<\/p>\r\n&nbsp;\r\n\r\n<strong>Example 3:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:\r\n<p style=\"text-align: center;\">\u00a0If P, then Q if and only if R.<\/p>\r\nHere the comma comes after the P, so \u201cQ if and only if R\u201d is inside parentheses. Using symbols that looks like:\r\n<p style=\"text-align: center;\">[latex]P\\rightarrow \\left(Q\\leftrightarrow R\\right)[\/latex].<\/p>\r\n&nbsp;\r\n\r\nNow let\u2019s look at examples using English statements.\r\n\r\n<strong>Example 4: Translate the English statement into logical symbols.<\/strong>\r\n\r\nLet P: The air is cold\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0Q: The humidity is high\u00a0\u00a0\u00a0\u00a0 R: I will wear shorts\r\n\r\nThe air is cold and the humidity is high, or I will wear shorts.\r\n\r\nSolution: [latex]\\left(P\\land Q\\right)\\lor{R}[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Example 5: Translate the English statement into logical symbols.<\/strong>\r\n\r\nLet \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza\r\n\r\nThe bookstore is not closed and the parking lot is not full, or the cafeteria is serving pizza.\r\n\r\nSolution: [latex]\\left({\\sim P}\\land \\sim Q\\right)\\lor{R}[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Example 6: Translate the English statement into logical symbols.<\/strong>\r\n\r\nLet \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza\r\n\r\nThe bookstore is not closed, and the parking lot is not full or the cafeteria is serving pizza.\r\n\r\nSolution: [latex]{\\sim P}\\land \\left( \\sim Q\\lor {R}\\right)[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Example 7: Translate the English statement into logical symbols.<\/strong>\r\n\r\nLet \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza\r\n\r\nIf the bookstore is closed, then the parking lot is full if and only if the cafeteria is serving pizza.\r\n\r\nSolution: [latex]P\\rightarrow \\left(Q \\leftrightarrow R\\right)[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Example 8: Translate the English statement into logical symbols.<\/strong>\r\n\r\nLet \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza\r\n\r\nIf the bookstore is closed then the parking lot is full, if and only if the cafeteria is serving pizza.\r\n\r\nSolution: [latex]\\left(P\\rightarrow Q \\right)\\leftrightarrow R[\/latex]<\/div>\r\n<strong>Building the Truth Table<\/strong>\r\n\r\nWe set up a table with all possible combinations of truth values for the simple statements that make up the compound statement.\u00a0 Then we build new columns, one at a time.\u00a0 If there are parenthesis in the compound statements, we find the truth value of those first.\u00a0 Obviously, the best way to ensure that the intended order is followed is to use parenthesis.\u00a0 If there are no parenthesis the negation would come first and then the connectives <em>and, or, if... then, if and only if.<\/em>\r\n\r\nTruth tables really become useful when analyzing more complex Boolean statements.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCreate a truth table for the statement [latex]A\\wedge\\sim\\left(B\\vee{C}\\right)[\/latex]\r\n[reveal-answer q=\"550652\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"550652\"]\r\n\r\nIt helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for a, b, and c. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we\u2019ll list the truth values for the innermost expression, [latex]B\\vee{C}[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>a<\/em><\/td>\r\n<td><em>b<\/em><\/td>\r\n<td><em>c<\/em><\/td>\r\n<td><em>b<\/em> \u22c1 c<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNext we can find the negation of [latex]b\\vee{c}[\/latex], working off the [latex]b\\vee{c}[\/latex] column we just created.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>a<\/em><\/td>\r\n<td><em>b<\/em><\/td>\r\n<td><em>c<\/em><\/td>\r\n<td>[latex]b\\vee{c}[\/latex]<\/td>\r\n<td>[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFinally, we find the values of a <em>and<\/em>\u00a0[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 18.9792px;\"><em>a<\/em><\/td>\r\n<td style=\"width: 18.9792px;\"><em>b<\/em><\/td>\r\n<td style=\"width: 19.6458px;\"><em>c<\/em><\/td>\r\n<td style=\"width: 121.646px;\">[latex]b\\vee{c}[\/latex]<\/td>\r\n<td style=\"width: 196.312px;\">[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]<\/td>\r\n<td style=\"width: 240.979px;\">[latex]a\\wedge\\sim\\left(b{\\vee}c\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 19.6458px;\">T<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">\u00a0F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 19.6458px;\">F<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 19.6458px;\">T<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 19.6458px;\">F<\/td>\r\n<td style=\"width: 121.646px;\">F<\/td>\r\n<td style=\"width: 196.312px;\">T<\/td>\r\n<td style=\"width: 240.979px;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 19.6458px;\">T<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 18.9792px;\">T<\/td>\r\n<td style=\"width: 19.6458px;\">F<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 19.6458px;\">T<\/td>\r\n<td style=\"width: 121.646px;\">T<\/td>\r\n<td style=\"width: 196.312px;\">F<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 18.9792px;\">F<\/td>\r\n<td style=\"width: 19.6458px;\">F<\/td>\r\n<td style=\"width: 121.646px;\">F<\/td>\r\n<td style=\"width: 196.312px;\">T<\/td>\r\n<td style=\"width: 240.979px;\">F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt turns out that this complex expression is only true in one case: if a is true, b is false, and c is false.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]25595[\/ohm_question]\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Commas in Logical Statements<\/li>\n<li>Order of Logical Operations\n<ul>\n<li>Create truth tables for complex statements using all operations (negations, conjunctions, disjunctions, conditionals, biconditionals)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>When combining operations in logic, commas are important.<\/p>\n<div class=\"textbox\">\n<p><strong>Commas in Logical Statements<\/strong><\/p>\n<p>When we translate English statements into logical symbols, <strong>commas are used to show where the parentheses are placed<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p>An easy way to visualize it is to think in terms of the symbols.<\/p>\n<p><strong>Example 1:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:<\/p>\n<p style=\"text-align: center;\">P and Q, or R.<\/p>\n<p>The comma comes after the Q, so \u201cP and Q\u201d is inside parentheses.<\/p>\n<p style=\"text-align: center;\">(P and Q), or R.<\/p>\n<p>Or, if we write using all symbols, P and Q, or R becomes:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(P\\land Q\\right)\\lor{R}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 2:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:<\/p>\n<p style=\"text-align: center;\">\u00a0If P then Q, if and only if R.<\/p>\n<p>The comma comes after the Q, so \u201cif P then Q\u201d is inside parentheses. Using symbols that looks like:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(P\\rightarrow Q\\right)\\leftrightarrow R[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 3:<\/strong> Suppose P, Q, and R are simple statements. Now consider the compound statement:<\/p>\n<p style=\"text-align: center;\">\u00a0If P, then Q if and only if R.<\/p>\n<p>Here the comma comes after the P, so \u201cQ if and only if R\u201d is inside parentheses. Using symbols that looks like:<\/p>\n<p style=\"text-align: center;\">[latex]P\\rightarrow \\left(Q\\leftrightarrow R\\right)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Now let\u2019s look at examples using English statements.<\/p>\n<p><strong>Example 4: Translate the English statement into logical symbols.<\/strong><\/p>\n<p>Let P: The air is cold\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0Q: The humidity is high\u00a0\u00a0\u00a0\u00a0 R: I will wear shorts<\/p>\n<p>The air is cold and the humidity is high, or I will wear shorts.<\/p>\n<p>Solution: [latex]\\left(P\\land Q\\right)\\lor{R}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 5: Translate the English statement into logical symbols.<\/strong><\/p>\n<p>Let \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza<\/p>\n<p>The bookstore is not closed and the parking lot is not full, or the cafeteria is serving pizza.<\/p>\n<p>Solution: [latex]\\left({\\sim P}\\land \\sim Q\\right)\\lor{R}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 6: Translate the English statement into logical symbols.<\/strong><\/p>\n<p>Let \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza<\/p>\n<p>The bookstore is not closed, and the parking lot is not full or the cafeteria is serving pizza.<\/p>\n<p>Solution: [latex]{\\sim P}\\land \\left( \\sim Q\\lor {R}\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 7: Translate the English statement into logical symbols.<\/strong><\/p>\n<p>Let \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza<\/p>\n<p>If the bookstore is closed, then the parking lot is full if and only if the cafeteria is serving pizza.<\/p>\n<p>Solution: [latex]P\\rightarrow \\left(Q \\leftrightarrow R\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 8: Translate the English statement into logical symbols.<\/strong><\/p>\n<p>Let \u00a0\u00a0\u00a0P: The bookstore is closed\u00a0\u00a0\u00a0\u00a0\u00a0 Q: The parking lot is full\u00a0\u00a0\u00a0\u00a0 R: the cafeteria is serving pizza<\/p>\n<p>If the bookstore is closed then the parking lot is full, if and only if the cafeteria is serving pizza.<\/p>\n<p>Solution: [latex]\\left(P\\rightarrow Q \\right)\\leftrightarrow R[\/latex]<\/p><\/div>\n<p><strong>Building the Truth Table<\/strong><\/p>\n<p>We set up a table with all possible combinations of truth values for the simple statements that make up the compound statement.\u00a0 Then we build new columns, one at a time.\u00a0 If there are parenthesis in the compound statements, we find the truth value of those first.\u00a0 Obviously, the best way to ensure that the intended order is followed is to use parenthesis.\u00a0 If there are no parenthesis the negation would come first and then the connectives <em>and, or, if&#8230; then, if and only if.<\/em><\/p>\n<p>Truth tables really become useful when analyzing more complex Boolean statements.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Create a truth table for the statement [latex]A\\wedge\\sim\\left(B\\vee{C}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q550652\">Show Solution<\/span><\/p>\n<div id=\"q550652\" class=\"hidden-answer\" style=\"display: none\">\n<p>It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for a, b, and c. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we\u2019ll list the truth values for the innermost expression, [latex]B\\vee{C}[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>a<\/em><\/td>\n<td><em>b<\/em><\/td>\n<td><em>c<\/em><\/td>\n<td><em>b<\/em> \u22c1 c<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Next we can find the negation of [latex]b\\vee{c}[\/latex], working off the [latex]b\\vee{c}[\/latex] column we just created.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>a<\/em><\/td>\n<td><em>b<\/em><\/td>\n<td><em>c<\/em><\/td>\n<td>[latex]b\\vee{c}[\/latex]<\/td>\n<td>[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Finally, we find the values of a <em>and<\/em>\u00a0[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 18.9792px;\"><em>a<\/em><\/td>\n<td style=\"width: 18.9792px;\"><em>b<\/em><\/td>\n<td style=\"width: 19.6458px;\"><em>c<\/em><\/td>\n<td style=\"width: 121.646px;\">[latex]b\\vee{c}[\/latex]<\/td>\n<td style=\"width: 196.312px;\">[latex]\\sim\\left(b\\vee{c}\\right)[\/latex]<\/td>\n<td style=\"width: 240.979px;\">[latex]a\\wedge\\sim\\left(b{\\vee}c\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 19.6458px;\">T<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">\u00a0F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 19.6458px;\">F<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 19.6458px;\">T<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 19.6458px;\">F<\/td>\n<td style=\"width: 121.646px;\">F<\/td>\n<td style=\"width: 196.312px;\">T<\/td>\n<td style=\"width: 240.979px;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 19.6458px;\">T<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 18.9792px;\">T<\/td>\n<td style=\"width: 19.6458px;\">F<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 19.6458px;\">T<\/td>\n<td style=\"width: 121.646px;\">T<\/td>\n<td style=\"width: 196.312px;\">F<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 18.9792px;\">F<\/td>\n<td style=\"width: 19.6458px;\">F<\/td>\n<td style=\"width: 121.646px;\">F<\/td>\n<td style=\"width: 196.312px;\">T<\/td>\n<td style=\"width: 240.979px;\">F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It turns out that this complex expression is only true in one case: if a is true, b is false, and c is false.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm25595\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25595&theme=oea&iframe_resize_id=ohm25595&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. 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