{"id":194,"date":"2023-02-08T20:00:18","date_gmt":"2023-02-08T20:00:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=194"},"modified":"2025-04-03T15:21:14","modified_gmt":"2025-04-03T15:21:14","slug":"set-theory-basics-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-background-youll-need-2\/","title":{"raw":"Set Theory and Logic: Background You'll Need  2","rendered":"Set Theory and Logic: Background You&#8217;ll Need  2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Simplify expressions using addition and multiplication<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Commutative Properties<\/h2>\r\n<p>The <strong>commutative properties<\/strong> are fundamental rules of arithmetic that apply to the operations of addition and multiplication. These properties assert that the order in which two numbers are added or multiplied does not change the result. Such properties are essential for understanding the flexibility we have when rearranging and simplifying expressions in mathematics.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>commutative properties<\/h3>\r\n<p><strong>Commutative Property of Addition<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>[latex]a+b=b+a[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><br \/>\r\n<strong>Commutative Property of Multiplication<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>[latex]a\\cdot b=b\\cdot a[\/latex]<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2872[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2873[\/ohm2_question]<\/section>\r\n<section>\r\n<h2>Associative Properties<\/h2>\r\n<p><strong>Associative properties<\/strong> refer to the way numbers can be grouped within parentheses when added or multiplied without affecting the overall sum or product. These properties highlight that regardless of how we pair numbers, the result remains the same, simplifying computation and providing a basis for more advanced algebraic concepts.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>associative properties<\/h3>\r\n<p><strong>Associative Property of Addition<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>[latex]\\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><br \/>\r\n<strong>Associative Property of Multiplication<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>[latex]\\left(a\\cdot b\\right)\\cdot c=a\\cdot \\left(b\\cdot c\\right)[\/latex]<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2874[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2875[\/ohm2_question]<\/section>\r\n<h2>Simplify Expressions Using the Commutative and Associative Properties<\/h2>\r\n<p>Understanding the commutative and associative properties can greatly simplify the process of solving mathematical expressions. The commutative property allows us to rearrange numbers in addition or multiplication without changing the result, offering flexibility in how we approach problems. Meanwhile, the associative property lets us regroup numbers\u2014also in addition or multiplication\u2014ensuring that no matter how we pair the numbers, the outcome is consistent. Mastering these foundational rules will streamline your problem-solving, making complex calculations more manageable.<\/p>\r\n<section class=\"textbox example\">Simplify: [latex]-84n+\\left(-73n\\right)+84n[\/latex]<br \/>\r\n[reveal-answer q=\"679824\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"679824\"]<br \/>\r\nNotice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.\r\n\r\n<table id=\"eip-id1168469525330\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Re-order the terms.<\/td>\r\n<td>[latex]-84n+84n+\\left(-73n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add left to right.<\/td>\r\n<td>[latex]0+\\left(-73n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]-73n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2876[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<br \/>\r\n[reveal-answer q=\"543511\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"543511\"]<br \/>\r\nNotice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.\r\n\r\n<table id=\"eip-id1168466660267\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Re-order the terms.<\/td>\r\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{15}{7}\\cdot \\frac{8}{23}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply left to right.<\/td>\r\n<td>[latex]1\\cdot\\Large\\frac{8}{23}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{8}{23}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2882[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<br \/>\r\n[reveal-answer q=\"964224\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"964224\"]<br \/>\r\nNotice that the sum of the second and third coefficients is a whole number.\r\n\r\n<table id=\"eip-id1168468331350\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change the grouping.<\/td>\r\n<td>[latex]6.47q+\\left(9.99q+1.01q\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add in the parentheses first.<\/td>\r\n<td>[latex]6.47q+\\left(11.00q\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]17.47q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2884[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Simplify: [latex]6\\left(9x\\right)[\/latex]<br \/>\r\n[reveal-answer q=\"587700\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"587700\"]\r\n\r\n<table id=\"eip-id1168467313641\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]6\\left(9x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the associative property of multiplication to re-group.<\/td>\r\n<td>[latex]\\left(6\\cdot 9\\right)x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply in the parentheses.<\/td>\r\n<td>[latex]54x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2886[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify expressions using addition and multiplication<\/li>\n<\/ul>\n<\/section>\n<h2>Commutative Properties<\/h2>\n<p>The <strong>commutative properties<\/strong> are fundamental rules of arithmetic that apply to the operations of addition and multiplication. These properties assert that the order in which two numbers are added or multiplied does not change the result. Such properties are essential for understanding the flexibility we have when rearranging and simplifying expressions in mathematics.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>commutative properties<\/h3>\n<p><strong>Commutative Property of Addition<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>\n<strong>Commutative Property of Multiplication<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2872\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2872&theme=lumen&iframe_resize_id=ohm2872&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2873\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2873&theme=lumen&iframe_resize_id=ohm2873&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Associative Properties<\/h2>\n<p><strong>Associative properties<\/strong> refer to the way numbers can be grouped within parentheses when added or multiplied without affecting the overall sum or product. These properties highlight that regardless of how we pair numbers, the result remains the same, simplifying computation and providing a basis for more advanced algebraic concepts.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>associative properties<\/h3>\n<p><strong>Associative Property of Addition<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>\n<strong>Associative Property of Multiplication<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\left(a\\cdot b\\right)\\cdot c=a\\cdot \\left(b\\cdot c\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2874\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2874&theme=lumen&iframe_resize_id=ohm2874&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2875\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2875&theme=lumen&iframe_resize_id=ohm2875&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Simplify Expressions Using the Commutative and Associative Properties<\/h2>\n<p>Understanding the commutative and associative properties can greatly simplify the process of solving mathematical expressions. The commutative property allows us to rearrange numbers in addition or multiplication without changing the result, offering flexibility in how we approach problems. Meanwhile, the associative property lets us regroup numbers\u2014also in addition or multiplication\u2014ensuring that no matter how we pair the numbers, the outcome is consistent. Mastering these foundational rules will streamline your problem-solving, making complex calculations more manageable.<\/p>\n<section class=\"textbox example\">Simplify: [latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q679824\">Show Solution<\/button><\/p>\n<div id=\"q679824\" class=\"hidden-answer\" style=\"display: none\">\nNotice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.<\/p>\n<table id=\"eip-id1168469525330\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Re-order the terms.<\/td>\n<td>[latex]-84n+84n+\\left(-73n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add left to right.<\/td>\n<td>[latex]0+\\left(-73n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]-73n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2876\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2876&theme=lumen&iframe_resize_id=ohm2876&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Simplify: [latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q543511\">Show Solution<\/button><\/p>\n<div id=\"q543511\" class=\"hidden-answer\" style=\"display: none\">\nNotice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.<\/p>\n<table id=\"eip-id1168466660267\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Re-order the terms.<\/td>\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{15}{7}\\cdot \\frac{8}{23}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply left to right.<\/td>\n<td>[latex]1\\cdot\\Large\\frac{8}{23}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{8}{23}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2882\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2882&theme=lumen&iframe_resize_id=ohm2882&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Simplify: [latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q964224\">Show Solution<\/button><\/p>\n<div id=\"q964224\" class=\"hidden-answer\" style=\"display: none\">\nNotice that the sum of the second and third coefficients is a whole number.<\/p>\n<table id=\"eip-id1168468331350\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change the grouping.<\/td>\n<td>[latex]6.47q+\\left(9.99q+1.01q\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the parentheses first.<\/td>\n<td>[latex]6.47q+\\left(11.00q\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]17.47q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2884\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2884&theme=lumen&iframe_resize_id=ohm2884&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Simplify: [latex]6\\left(9x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q587700\">Show Solution<\/button><\/p>\n<div id=\"q587700\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467313641\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]6\\left(9x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the associative property of multiplication to re-group.<\/td>\n<td>[latex]\\left(6\\cdot 9\\right)x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply in the parentheses.<\/td>\n<td>[latex]54x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2886\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2886&theme=lumen&iframe_resize_id=ohm2886&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"background_you_need","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Download for free at 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