{"id":6780,"date":"2021-12-30T21:29:18","date_gmt":"2021-12-30T21:29:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6780"},"modified":"2022-01-06T22:14:53","modified_gmt":"2022-01-06T22:14:53","slug":"1-2-subsets","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-2-subsets\/","title":{"raw":"1.2 Subsets","rendered":"1.2 Subsets"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nThe basics of subsets\r\n<ul>\r\n \t<li>Number of distinct subsets<\/li>\r\n \t<li>The basics of proper subsets<\/li>\r\n \t<li>Number of distinct proper subsets<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Subsets<\/h2>\r\nSometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Madonna albums.\r\n<div class=\"textbox examples\">\r\n<h3>Subsets of real numbers<\/h3>\r\nThe idea of subsets can also be applied to the sets of real numbers you studied previously. For example, the set of all whole numbers is a subset of the set of all integers. The set of integers in turn is contained within the set of rational numbers.\r\n\r\nWe say <em>the integers are a subset of the rational numbers<\/em>. You'll see below in fact that the integers are a <em>proper subset<\/em> of the rational numbers.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Subset<\/h3>\r\nA <strong>subset<\/strong> of a set <em>A<\/em> is another set that contains only elements from the set <em>A<\/em>, but may not contain all the elements of <em>A<\/em>.\r\n\r\nIf <em>B<\/em> is a subset of <em>A<\/em>, we write <em>B<\/em> \u2286 <em>A<\/em>\r\n\r\nA <strong>proper subset<\/strong> is a subset that is not identical to the original set\u2014it contains fewer elements.\r\n\r\nIf <em>B<\/em> is a proper subset of <em>A<\/em>, we write <em>B<\/em> \u2282 <em>A<\/em>\r\n\r\nNote: a subset contains more elements than a proper subset. Therefore, thinking of the line underneath as an equivalence symbol can be helpful.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider these three sets:\r\n\r\n<em>A<\/em> = the set of all even numbers\r\n<em>B<\/em> = {2, 4, 6}\r\n<em>C<\/em> = {2, 3, 4, 6}\r\n\r\nHere <em>B<\/em> \u2282 <em>A<\/em> since every element of <em>B<\/em> is also an even number, so is an element of <em>A<\/em>.\r\n\r\nMore formally, we could say <em>B<\/em> \u2282 <em>A<\/em> since if <em>x <\/em>\u2208\u00a0<em>B<\/em>, then <em>x <\/em>\u2208 <em>A<\/em>.\r\n\r\nIt is also true that <em>B<\/em> \u2282 <em>C<\/em>.\r\n\r\n<em>C<\/em> is not a subset of <em>A<\/em>, since C contains an element, 3, that is not contained in <em>A<\/em>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5xthPHH4i_A?list=PL7138FAEC01D6F3F3\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose a set contains the plays \u201cMuch Ado About Nothing,\u201d \u201cMacBeth,\u201d and \u201cA Midsummer\u2019s Night Dream.\u201d What is a larger set this might be a subset of?\r\n[reveal-answer q=\"42047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42047\"]There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\} [\/latex]. Which of the following sets is [latex]A [\/latex] a subset of?\r\n[latex]X = \\{1, 3, 7, 5\\} [\/latex]\r\n[latex]Y = \\{1, 3 \\} [\/latex]\r\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]\r\n[reveal-answer q=\"3546\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3546\"] [latex] X [\/latex] and [latex] Z [\/latex] [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nGiven the\u00a0set: <em>A<\/em> = {<em>a<\/em>, <em>b<\/em>, <em>c<\/em>, <em>d<\/em>}. List all of the subsets of <em>A\r\n<\/em>[reveal-answer q=\"706217\"]Show Solution[\/reveal-answer]<em>\r\n<\/em>[hidden-answer a=\"706217\"]{} (or \u00d8), {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d},\u00a0{b,c,d}, {a,b,c,d}\r\n\r\nYou can see that there are 16 subsets, 15 of which are proper subsets.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nListing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn the previous example, there are four elements. For the first element, <em>a<\/em>, either it\u2019s in the set or it\u2019s not. Thus there are 2 choices for that first element. Similarly, there are two choices for <em>b<\/em>\u2014either it\u2019s in the set or it\u2019s not. Using just those two elements,\u00a0list all the possible subsets of the set {a,b}\r\n[reveal-answer q=\"857946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"857946\"]\r\n\r\n{}\u2014both elements are not in the set\r\n{<em>a<\/em>}\u2014<em>a<\/em> is in;\u00a0<em>b<\/em> is not in the set\r\n{<em>b<\/em>}\u2014<em>a<\/em> is not in the set;\u00a0<em>b<\/em> is in\r\n{<em>a<\/em>,<em>b<\/em>}\u2014<em>a<\/em> is in; <em>b<\/em> is in\r\n\r\nTwo choices for <em>a<\/em>\u00a0times the two for <em>b<\/em>\u00a0gives us [latex]2^{2}=4[\/latex] subsets.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall exponential notation<\/h3>\r\n<div>Recall that the expression [latex]a^{m}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]m[\/latex] times.<\/div>\r\n<div><\/div>\r\n<div>Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/div>\r\n<\/div>\r\nNow let\u2019s include <em>c,\u00a0<\/em>just for fun. List all the possible subsets of the new set {a,b,c}.\r\nAgain, either <em>c<\/em> is included or it isn\u2019t, which gives us two choices. The outcomes are {}, {<em>a<\/em>}, {<em>b<\/em>}, {<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>}, {<em>a<\/em>,<em>c<\/em>}, {<em>b<\/em>,<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>}. Note that there are [latex]2^{3}=8[\/latex] subsets.\r\n\r\nIf you include\u00a0four elements, there would be [latex]2^{4}=16[\/latex] subsets. 15 of those subsets are proper, 1 subset, namely {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>,<em>d<\/em>}, is not.\r\n<div class=\"textbox\">In general, if you have <em>n<\/em> elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Number of Subsets<\/h3>\r\n<div><\/div>\r\n<div>\r\n\r\n<strong>Number of Subsets<\/strong>\r\nFirst, subsets are smaller collections of a sets\u2019 elements.\r\n\r\nFor example, suppose set A = {red, yellow, blue}. To find all the subsets, just list all the possible combinations of elements:\r\n1. { } (This is the empty set. <strong>The empty set is a subset of all sets.<\/strong>)\r\n2. {red}\r\n3. {yellow}\r\n4. {blue}\r\n5. {red, yellow}\r\n6. {red, blue}\r\n7. {yellow, blue}\r\n8. {red, yellow, blue} (Yes, the set can be a subset of itself!)\r\n\r\nNotice there were 8 possible combinations of elements, including the empty set. It turns out that all sets with 3 elements have 8 possible combinations of elements.\r\n\r\nHere\u2019s the pattern on how you figure out the number of distinct subsets that a set has:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A set with this many elements\u2026<\/td>\r\n<td>Has this many distinct subsets\u2026<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>2\u00a0\u00a0 (this is 2<sup>1 <\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>4\u00a0\u00a0 (this is 2<sup>2 <\/sup>\u00a0or 2 x 2)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>8\u00a0\u00a0 (this is 2<sup>3 <\/sup>\u00a0or 2 x 2 x 2)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>16\u00a0\u00a0 (this is 2<sup>4 <\/sup>\u00a0or 2 x 2 x 2 x 2)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>n, where is any whole number<\/td>\r\n<td>2<sup>n <\/sup>\u00a0\u00a0 (you can use the exponent key on your calculator to find 2<sup>n <\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>The number of distinct subsets of a finite set A is 2<sup>n <\/sup><\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Number of Distinct Proper Subsets<\/h3>\r\nAn important fact to remember about proper subsets is that every set is a subset of itself, HOWEVER, no set is a proper subset of itself.\r\n\r\nConsider, again, the set A = {red, yellow, blue}. To find all the possible proper subsets, just list all the possible combinations of elements, but do NOT include the set itself:\r\n1. { } (This is the empty set. The empty set is a subset of all sets.)\r\n2. {red}\r\n3. {yellow}\r\n4. {blue}\r\n5. {red, yellow}\r\n6. {red, blue}\r\n7. {yellow, blue}\r\n\r\nNotice that {red, yellow, blue} is not listed in the proper subsets. So, to find the number of distinct proper subsets subtract 1 from the number of distinct subsets. <strong>So, the number of distinct proper subsets of a finite set A is 2<sup>n <\/sup> - 1<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nSuppose you went out for pizza. There were only four possible toppings: pepperoni, sausage, olives, and mushrooms. How many different variations of pizza are there?\r\n\r\n[reveal-answer q=\"990937\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"990937\"]Answer = Since there are 4 possible toppings, the exponent is 4. So, there are 2<sup>4 <\/sup> = 16 possible variations of pizza.\r\n\r\nHere is the list of all subsets:\r\n1. No toppings (this is like the empty set\u2026no toppings)\r\n2. P (for pepperoni)\r\n3. S (for sausage)\r\n4. O (for olives)\r\n5. M (for mushrooms)\r\n6. P and S\r\n7. P and O\r\n8. P and M\r\n9. S and O\r\n10. S and M\r\n11. O and M\r\n12. P, S, and O\r\n13. P, S, and M\r\n14. P, O, and M\r\n15. S, O, and M\r\n16. P, S, O, and M\r\n\r\nIf we wanted the number of distinct proper subsets, we would subtract the last one (P, S, O, and M). Maybe there was a rule at the pizza place that said you could not have all four toppings on one pizza. So, then there would only be 15 possible combinations of toppings..[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>YOU TRY<\/h3>\r\nSuppose you went to an all-you-can-eat buffet with the following items: cashew chicken (CC), shrimp fried rice (SFR), sweet and sour pork (SSP), broccoli with garlic sauce (BGS), and egg rolls (ER).\r\n\r\na) Using the appropriate formula, show how you could calculate the number of variations on your plate without listing them out.\r\n\r\n[reveal-answer q=\"630980\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"630980\"]There are 5 food items, therefore there are 2<sup>5\u00a0<\/sup> or 32 possible variations.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nb) List the possible variations that you could put on your plate.\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"7245\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"7245\"]\r\n\r\n1. { }\r\n\r\n2. {CC}\r\n\r\n3. {SFR}\r\n\r\n4. {SSP}\r\n\r\n5. {BGS}\r\n\r\n6. {ER}\r\n\r\n7. {CC, SFR}\r\n\r\n8. {CC, SSP}\r\n\r\n9. {CC, BGS}\r\n\r\n10. {CC, ER}\r\n\r\n11. {SFR, SSP}\r\n\r\n12. {SFR, BGS}\r\n\r\n13. {SFR, ER}\r\n\r\n14. {SSP, BGS}\r\n\r\n15. {SSP, ER}\r\n\r\n16. {BGS, ER}\r\n\r\n17. {CC, SFR, SSP}\r\n\r\n18. {CC, SFR, BGS}\r\n\r\n19. {CC, SFR, ER}\r\n\r\n20. {CC, SSP, BGS}\r\n\r\n21. {CC, SSP, ER}\r\n\r\n22. {CC, BGS, ER}\r\n\r\n23. {SFR, SSP, BGS}\r\n\r\n24. {SFR, SSP, ER}\r\n\r\n25. {SFR, BGS, ER}\r\n\r\n26. {SSP, BGS, ER}\r\n\r\n27. {CC, SFR, SSP, BGS}\r\n\r\n28. {CC, SFR, SSP, ER}\r\n\r\n29. {CC, SFR, BGS, ER}\r\n\r\n30. {CC, SSP, BGS, ER}\r\n\r\n31. (SFR, SSP, BGS, ER}\r\n\r\n32. {CC, SFR, SSP, BGS, ER}[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nc) Would these variations be considered subsets or proper subsets, and why?\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"488497\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"488497\"]They are considered subsets because of the last variation, the one with all five food items. A proper subset cannot contain itself, but subsets can.[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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<p>The basics of subsets<\/p>\n<ul>\n<li>Number of distinct subsets<\/li>\n<li>The basics of proper subsets<\/li>\n<li>Number of distinct proper subsets<\/li>\n<\/ul>\n<\/div>\n<h2>Subsets<\/h2>\n<p>Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Madonna albums.<\/p>\n<div class=\"textbox examples\">\n<h3>Subsets of real numbers<\/h3>\n<p>The idea of subsets can also be applied to the sets of real numbers you studied previously. For example, the set of all whole numbers is a subset of the set of all integers. The set of integers in turn is contained within the set of rational numbers.<\/p>\n<p>We say <em>the integers are a subset of the rational numbers<\/em>. You&#8217;ll see below in fact that the integers are a <em>proper subset<\/em> of the rational numbers.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Subset<\/h3>\n<p>A <strong>subset<\/strong> of a set <em>A<\/em> is another set that contains only elements from the set <em>A<\/em>, but may not contain all the elements of <em>A<\/em>.<\/p>\n<p>If <em>B<\/em> is a subset of <em>A<\/em>, we write <em>B<\/em> \u2286 <em>A<\/em><\/p>\n<p>A <strong>proper subset<\/strong> is a subset that is not identical to the original set\u2014it contains fewer elements.<\/p>\n<p>If <em>B<\/em> is a proper subset of <em>A<\/em>, we write <em>B<\/em> \u2282 <em>A<\/em><\/p>\n<p>Note: a subset contains more elements than a proper subset. Therefore, thinking of the line underneath as an equivalence symbol can be helpful.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider these three sets:<\/p>\n<p><em>A<\/em> = the set of all even numbers<br \/>\n<em>B<\/em> = {2, 4, 6}<br \/>\n<em>C<\/em> = {2, 3, 4, 6}<\/p>\n<p>Here <em>B<\/em> \u2282 <em>A<\/em> since every element of <em>B<\/em> is also an even number, so is an element of <em>A<\/em>.<\/p>\n<p>More formally, we could say <em>B<\/em> \u2282 <em>A<\/em> since if <em>x <\/em>\u2208\u00a0<em>B<\/em>, then <em>x <\/em>\u2208 <em>A<\/em>.<\/p>\n<p>It is also true that <em>B<\/em> \u2282 <em>C<\/em>.<\/p>\n<p><em>C<\/em> is not a subset of <em>A<\/em>, since C contains an element, 3, that is not contained in <em>A<\/em><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Sets: basics\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5xthPHH4i_A?list=PL7138FAEC01D6F3F3\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose a set contains the plays \u201cMuch Ado About Nothing,\u201d \u201cMacBeth,\u201d and \u201cA Midsummer\u2019s Night Dream.\u201d What is a larger set this might be a subset of?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42047\">Show Solution<\/span><\/p>\n<div id=\"q42047\" class=\"hidden-answer\" style=\"display: none\">There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\}[\/latex]. Which of the following sets is [latex]A[\/latex] a subset of?<br \/>\n[latex]X = \\{1, 3, 7, 5\\}[\/latex]<br \/>\n[latex]Y = \\{1, 3 \\}[\/latex]<br \/>\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q3546\">Show Solution<\/span><\/p>\n<div id=\"q3546\" class=\"hidden-answer\" style=\"display: none\"> [latex]X[\/latex] and [latex]Z[\/latex] <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Given the\u00a0set: <em>A<\/em> = {<em>a<\/em>, <em>b<\/em>, <em>c<\/em>, <em>d<\/em>}. List all of the subsets of <em>A<br \/>\n<\/em><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706217\">Show Solution<\/span><em><br \/>\n<\/em><\/p>\n<div id=\"q706217\" class=\"hidden-answer\" style=\"display: none\">{} (or \u00d8), {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d},\u00a0{b,c,d}, {a,b,c,d}<\/p>\n<p>You can see that there are 16 subsets, 15 of which are proper subsets.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Listing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In the previous example, there are four elements. For the first element, <em>a<\/em>, either it\u2019s in the set or it\u2019s not. Thus there are 2 choices for that first element. Similarly, there are two choices for <em>b<\/em>\u2014either it\u2019s in the set or it\u2019s not. Using just those two elements,\u00a0list all the possible subsets of the set {a,b}<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q857946\">Show Solution<\/span><\/p>\n<div id=\"q857946\" class=\"hidden-answer\" style=\"display: none\">\n<p>{}\u2014both elements are not in the set<br \/>\n{<em>a<\/em>}\u2014<em>a<\/em> is in;\u00a0<em>b<\/em> is not in the set<br \/>\n{<em>b<\/em>}\u2014<em>a<\/em> is not in the set;\u00a0<em>b<\/em> is in<br \/>\n{<em>a<\/em>,<em>b<\/em>}\u2014<em>a<\/em> is in; <em>b<\/em> is in<\/p>\n<p>Two choices for <em>a<\/em>\u00a0times the two for <em>b<\/em>\u00a0gives us [latex]2^{2}=4[\/latex] subsets.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall exponential notation<\/h3>\n<div>Recall that the expression [latex]a^{m}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]m[\/latex] times.<\/div>\n<div><\/div>\n<div>Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/div>\n<\/div>\n<p>Now let\u2019s include <em>c,\u00a0<\/em>just for fun. List all the possible subsets of the new set {a,b,c}.<br \/>\nAgain, either <em>c<\/em> is included or it isn\u2019t, which gives us two choices. The outcomes are {}, {<em>a<\/em>}, {<em>b<\/em>}, {<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>}, {<em>a<\/em>,<em>c<\/em>}, {<em>b<\/em>,<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>}. Note that there are [latex]2^{3}=8[\/latex] subsets.<\/p>\n<p>If you include\u00a0four elements, there would be [latex]2^{4}=16[\/latex] subsets. 15 of those subsets are proper, 1 subset, namely {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>,<em>d<\/em>}, is not.<\/p>\n<div class=\"textbox\">In general, if you have <em>n<\/em> elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.<\/div>\n<div class=\"textbox examples\">\n<h3>Number of Subsets<\/h3>\n<div><\/div>\n<div>\n<p><strong>Number of Subsets<\/strong><br \/>\nFirst, subsets are smaller collections of a sets\u2019 elements.<\/p>\n<p>For example, suppose set A = {red, yellow, blue}. To find all the subsets, just list all the possible combinations of elements:<br \/>\n1. { } (This is the empty set. <strong>The empty set is a subset of all sets.<\/strong>)<br \/>\n2. {red}<br \/>\n3. {yellow}<br \/>\n4. {blue}<br \/>\n5. {red, yellow}<br \/>\n6. {red, blue}<br \/>\n7. {yellow, blue}<br \/>\n8. {red, yellow, blue} (Yes, the set can be a subset of itself!)<\/p>\n<p>Notice there were 8 possible combinations of elements, including the empty set. It turns out that all sets with 3 elements have 8 possible combinations of elements.<\/p>\n<p>Here\u2019s the pattern on how you figure out the number of distinct subsets that a set has:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A set with this many elements\u2026<\/td>\n<td>Has this many distinct subsets\u2026<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2\u00a0\u00a0 (this is 2<sup>1 <\/sup>)<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4\u00a0\u00a0 (this is 2<sup>2 <\/sup>\u00a0or 2 x 2)<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>8\u00a0\u00a0 (this is 2<sup>3 <\/sup>\u00a0or 2 x 2 x 2)<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>16\u00a0\u00a0 (this is 2<sup>4 <\/sup>\u00a0or 2 x 2 x 2 x 2)<\/td>\n<\/tr>\n<tr>\n<td>n, where is any whole number<\/td>\n<td>2<sup>n <\/sup>\u00a0\u00a0 (you can use the exponent key on your calculator to find 2<sup>n <\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>The number of distinct subsets of a finite set A is 2<sup>n <\/sup><\/strong>.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Number of Distinct Proper Subsets<\/h3>\n<p>An important fact to remember about proper subsets is that every set is a subset of itself, HOWEVER, no set is a proper subset of itself.<\/p>\n<p>Consider, again, the set A = {red, yellow, blue}. To find all the possible proper subsets, just list all the possible combinations of elements, but do NOT include the set itself:<br \/>\n1. { } (This is the empty set. The empty set is a subset of all sets.)<br \/>\n2. {red}<br \/>\n3. {yellow}<br \/>\n4. {blue}<br \/>\n5. {red, yellow}<br \/>\n6. {red, blue}<br \/>\n7. {yellow, blue}<\/p>\n<p>Notice that {red, yellow, blue} is not listed in the proper subsets. So, to find the number of distinct proper subsets subtract 1 from the number of distinct subsets. <strong>So, the number of distinct proper subsets of a finite set A is 2<sup>n <\/sup> &#8211; 1<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Suppose you went out for pizza. There were only four possible toppings: pepperoni, sausage, olives, and mushrooms. How many different variations of pizza are there?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990937\">Show Answer<\/span><\/p>\n<div id=\"q990937\" class=\"hidden-answer\" style=\"display: none\">Answer = Since there are 4 possible toppings, the exponent is 4. So, there are 2<sup>4 <\/sup> = 16 possible variations of pizza.<\/p>\n<p>Here is the list of all subsets:<br \/>\n1. No toppings (this is like the empty set\u2026no toppings)<br \/>\n2. P (for pepperoni)<br \/>\n3. S (for sausage)<br \/>\n4. O (for olives)<br \/>\n5. M (for mushrooms)<br \/>\n6. P and S<br \/>\n7. P and O<br \/>\n8. P and M<br \/>\n9. S and O<br \/>\n10. S and M<br \/>\n11. O and M<br \/>\n12. P, S, and O<br \/>\n13. P, S, and M<br \/>\n14. P, O, and M<br \/>\n15. S, O, and M<br \/>\n16. P, S, O, and M<\/p>\n<p>If we wanted the number of distinct proper subsets, we would subtract the last one (P, S, O, and M). Maybe there was a rule at the pizza place that said you could not have all four toppings on one pizza. So, then there would only be 15 possible combinations of toppings..<\/p><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>YOU TRY<\/h3>\n<p>Suppose you went to an all-you-can-eat buffet with the following items: cashew chicken (CC), shrimp fried rice (SFR), sweet and sour pork (SSP), broccoli with garlic sauce (BGS), and egg rolls (ER).<\/p>\n<p>a) Using the appropriate formula, show how you could calculate the number of variations on your plate without listing them out.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q630980\">Show Answer<\/span><\/p>\n<div id=\"q630980\" class=\"hidden-answer\" style=\"display: none\">There are 5 food items, therefore there are 2<sup>5\u00a0<\/sup> or 32 possible variations.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>b) List the possible variations that you could put on your plate.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q7245\">Show Answer<\/span><\/p>\n<div id=\"q7245\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. { }<\/p>\n<p>2. {CC}<\/p>\n<p>3. {SFR}<\/p>\n<p>4. {SSP}<\/p>\n<p>5. {BGS}<\/p>\n<p>6. {ER}<\/p>\n<p>7. {CC, SFR}<\/p>\n<p>8. {CC, SSP}<\/p>\n<p>9. {CC, BGS}<\/p>\n<p>10. {CC, ER}<\/p>\n<p>11. {SFR, SSP}<\/p>\n<p>12. {SFR, BGS}<\/p>\n<p>13. {SFR, ER}<\/p>\n<p>14. {SSP, BGS}<\/p>\n<p>15. {SSP, ER}<\/p>\n<p>16. {BGS, ER}<\/p>\n<p>17. {CC, SFR, SSP}<\/p>\n<p>18. {CC, SFR, BGS}<\/p>\n<p>19. {CC, SFR, ER}<\/p>\n<p>20. {CC, SSP, BGS}<\/p>\n<p>21. {CC, SSP, ER}<\/p>\n<p>22. {CC, BGS, ER}<\/p>\n<p>23. {SFR, SSP, BGS}<\/p>\n<p>24. {SFR, SSP, ER}<\/p>\n<p>25. {SFR, BGS, ER}<\/p>\n<p>26. {SSP, BGS, ER}<\/p>\n<p>27. {CC, SFR, SSP, BGS}<\/p>\n<p>28. {CC, SFR, SSP, ER}<\/p>\n<p>29. {CC, SFR, BGS, ER}<\/p>\n<p>30. {CC, SSP, BGS, ER}<\/p>\n<p>31. (SFR, SSP, BGS, ER}<\/p>\n<p>32. {CC, SFR, SSP, BGS, ER}<\/p><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>c) Would these variations be considered subsets or proper subsets, and why?<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q488497\">Show Answer<\/span><\/p>\n<div id=\"q488497\" class=\"hidden-answer\" style=\"display: none\">They are considered subsets because of the last variation, the one with all five food items. A proper subset cannot contain itself, but subsets can.<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. 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