{"id":6719,"date":"2020-10-08T15:39:03","date_gmt":"2020-10-08T15:39:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6719"},"modified":"2026-01-13T06:41:20","modified_gmt":"2026-01-13T06:41:20","slug":"2-3-describing-sets-as-intersections-or-unions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/2-3-describing-sets-as-intersections-or-unions\/","title":{"raw":"2.3: Describing Sets as Intersections or Unions","rendered":"2.3: Describing Sets as Intersections or Unions"},"content":{"raw":"
\r\n

section 2.3 Learning Objectives<\/h3>\r\n2.3:\u00a0 Describing Sets as Intersections or Unions<\/strong>\r\n
    \r\n \t
  • Find the intersection and union of two sets of numbers<\/li>\r\n \t
  • Use interval notation to describe sets of numbers as intersections and unions<\/li>\r\n \t
  • Recognize when an intersection has no solution or when a union has all real numbers as the solution<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Introduction to Sets<\/h2>\r\nA set is a collection of objects which are called elements<\/span>. The elements are distinct and, thus, are <\/span>listed only once. One way to identify a set is by the List or Roster Method<\/span>, whereby the elements <\/span>are listed within braces { }<\/span>. <\/span>\r\n\r\nFor example, consider the set of whole number less than 8. Using the List (Roster) Method, we would write this set as\r\n

    [latex]\\{0,1,2,3,4,5,6,7\\}[\/latex]<\/p>\r\nIt is worth noting that with the List Method, the elements in the set have no inherent order.\u00a0 So, for example, [latex]\\{4,6,0,2,1,7,3,5\\}[\/latex] would denote the same set.\r\n\r\nTo indicate <\/span>that <\/span>an object\u00a0<\/span>is an element of a set, the symbol [latex]\\in[\/latex]<\/span>\u00a0<\/span>is used.\u00a0<\/span>To indicate that an object is not an element of a set, the symbol [latex]\\notin[\/latex]<\/span>\u00a0<\/span>is used. Hence, in our above example, we could write [latex]5 \\in\\{0,1,2,3,4,5,6,7\\}[\/latex], whereas [latex]8 \\notin\\{0,1,2,3,4,5,6,7\\}[\/latex].<\/span>\r\n

    \r\n

    Example 1<\/h3>\r\nWrite <\/span>the first four\u00a0<\/span>months of a year as set M <\/span>using the List Method,\u00a0<\/span>identify one of <\/span>the months\u00a0<\/span>as <\/span>an element of the set<\/span>, and identify a month that is not an element of the set<\/span>.<\/span>\r\n\r\n[reveal-answer q=\"467869\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"467869\"]\u00a0M= {January, February, March, April}\r\nFebruary [latex]\\in[\/latex] M\r\nJuly [latex]\\notin[\/latex] M\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA set that has no elements i<\/span>s called an empty set and is denoted by the <\/span>symbol <\/span>\u00d8<\/span>.\u00a0<\/span>\r\n

    Find the intersection and union of sets<\/h2>\r\nSets can be joined together using the intersection of sets or the union of sets.\u00a0<\/span>\r\n
    \r\n

    The intersection <\/span><\/span>of two sets A and B is the set of all elements that are common to both A and B<\/span> <\/span>and is denoted as <\/span>A\u00a0<\/span>\u2229\u00a0<\/span>B.<\/span><\/strong><\/p>\r\n \r\n

    The union<\/span> of two sets A and B is the set of all elements in A or B<\/span> and is denoted as <\/span>A <\/span>\u222a\u00a0<\/span>B<\/span><\/strong><\/p>\r\n\r\n<\/div>\r\n \r\n

    \r\n

    Example 2<\/h3>\r\nFind the intersection: {2, 3, 4, 5, 6, 7} <\/span>\u2229\u00a0<\/span>{1, 2, 5, 7, 8, 9}\r\n<\/span>\r\n\r\n[reveal-answer q=\"347462\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"347462\"]{2, 5, 7}<\/span>\r\n\r\nSince elements 2, 5 and 7 were common to both sets, they are the elements in the intersection.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    Example 3<\/h3>\r\nFind the union: <\/span>{2, 4, 6, 8} <\/span>\u222a\u00a0<\/span>{2, 3, 5, 7}<\/span>\r\n\r\n[reveal-answer q=\"494868\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"494868\"]{2, 3, 4, 5, 6, 7, 8}<\/span>\r\n\r\nTo create the union between the two sets, we will take all elements that are in either the first set or the second set. Notice, even though the 2 was in both sets, we only write it once in the union.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    Use interval notation to describe sets of numbers as intersections and unions<\/h2>\r\nWhen two inequalities are joined by the word and<\/i>, the solution of the compound inequality occurs when both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or<\/i>, the solution of the compound inequality occurs when either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.\r\n\r\nIn this module we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.\r\n\r\nVenn diagrams use the concept of intersections and unions to show how much two or more things share in common. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently Cecilia has both of these qualities; therefore she is the intersection of the two.\r\n\r\n\"Two\r\n\r\nIn mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers x<\/em> can be, and what would the interval look like?\r\n\r\nIn words, x<\/em> must be less than 6 and at the same time,<\/em> it must be greater than 2, much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Let's look at a graph to see what numbers are possible with these constraints.\r\n\r\n\"Number\r\n\r\nThe numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a bounded<\/em> inequality and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. x<\/em> must be less than 6 and greater than two\u2014the values for x<\/em> will fall between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:\r\n\r\n\"Number\r\n\r\nOn the other hand, if you need to represent two things that\u00a0don't share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements but are often considered in the same application, such as online shopping or banking.\r\n\r\n\"Two\r\n\r\nIn mathematical terms, for example, [latex]x>6[\/latex]\u00a0or<\/em>\u00a0[latex]x<2[\/latex] is an inequality joined by the word \"or\". Using interval notation, we can describe each of these inequalities separately:<\/em><\/em>\r\n\r\n[latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x<2[\/latex] is the same as\u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0or<\/em> real numbers greater than 6. Can you see why we need to write them as two separate intervals? Let's look at a graph to get a clear picture of what is going on.\r\n\r\n\"Number\r\n\r\nWhen you place both of these inequalities on a graph, we can see that they share no numbers in common. This is what we call a union, as mentioned above. The interval notation associated with a union is a big U, so instead of writing or<\/em>, we join our intervals with a big U, like this:\r\n\r\n[latex]\\left(-\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]\r\n\r\n \r\n
    \r\n

    Example 4<\/h3>\r\nDraw the graph of the compound inequality [latex]x\\gt3[\/latex] or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"641470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641470\"]\r\n\r\nThe graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on 3 and a blue arrow drawn to the right to contain all the numbers greater than 3.\r\n\r\n\"Number\r\n\r\nThe graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than 4.\r\n\r\n\"Number\r\n\r\nWhat do you notice about the graph that combines these two inequalities?\r\n\r\n\"Number\r\n\r\nSince this compound inequality is an or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than 3 and less than or equal to 4 is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] or [latex]x\\le4[\/latex] is the set of all real numbers, and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex]\r\n\r\nA final graph that can be drawn to represent the solution is shown below:\r\n\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see two examples of how to express inequalities involving OR graphically and as an interval.\r\nhttps:\/\/youtu.be\/nKarzhZOFIk\r\n
    \r\n

    Example 5<\/h3>\r\nDraw a graph of the compound inequality:\u00a0[latex]x\\lt5[\/latex]\u00a0and<\/strong><\/em>\u00a0[latex]x\\ge\u22121[\/latex], and describe the set of x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"394627\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394627\"]\r\n\r\nThe graph of each individual inequality is shown in color.\r\n\r\n\"Number\r\n\r\nSince the word and <\/i>joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.\r\n\r\nThe solution to this compound inequality is shown below.\r\n\r\n\"Number\r\n\r\nNotice that this is a bounded inequality. You can rewrite [latex]x\\ge\u22121\\,\\text{and }x\\le5[\/latex] as [latex]\u22121\\le x\\le 5[\/latex]\u00a0since the solution is between [latex]\u22121[\/latex] and 5, including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex]\u00a0as \u201cx<\/i> is greater than or equal to [latex]\u22121[\/latex]\u00a0and<\/i> less than 5.\u201d You can rewrite an and<\/i> statement this way only if the answer is between<\/i> two numbers. The set of solutions to this inequality can be written in interval notation like this: [latex]\\left[{-1},{5}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    Example 6<\/h3>\r\nConsidering the compound inequality [latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex], describe the set of x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"870500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"870500\"]\r\n\r\nFirst, we can draw a graph to help us visualize the intervals. We are looking for values for x<\/em> that will satisfy both\u00a0<\/strong>inequalities since they are joined with the word and<\/em>.\r\n\r\n\"Number\r\n\r\nIn this case, there are no shared x<\/em>-values, and therefore there is no intersection for these two inequalities. We can write \"no solution,\" or DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video presents two examples of how to draw inequalities involving AND, as well as write the corresponding intervals.\r\n\r\nhttps:\/\/youtu.be\/LP3fsZNjJkc","rendered":"
    \n

    section 2.3 Learning Objectives<\/h3>\n

    2.3:\u00a0 Describing Sets as Intersections or Unions<\/strong><\/p>\n

      \n
    • Find the intersection and union of two sets of numbers<\/li>\n
    • Use interval notation to describe sets of numbers as intersections and unions<\/li>\n
    • Recognize when an intersection has no solution or when a union has all real numbers as the solution<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      Introduction to Sets<\/h2>\n

      A set is a collection of objects which are called elements<\/span>. The elements are distinct and, thus, are <\/span>listed only once. One way to identify a set is by the List or Roster Method<\/span>, whereby the elements <\/span>are listed within braces { }<\/span>. <\/span><\/p>\n

      For example, consider the set of whole number less than 8. Using the List (Roster) Method, we would write this set as<\/p>\n

      [latex]\\{0,1,2,3,4,5,6,7\\}[\/latex]<\/p>\n

      It is worth noting that with the List Method, the elements in the set have no inherent order.\u00a0 So, for example, [latex]\\{4,6,0,2,1,7,3,5\\}[\/latex] would denote the same set.<\/p>\n

      To indicate <\/span>that <\/span>an object\u00a0<\/span>is an element of a set, the symbol [latex]\\in[\/latex]<\/span>\u00a0<\/span>is used.\u00a0<\/span>To indicate that an object is not an element of a set, the symbol [latex]\\notin[\/latex]<\/span>\u00a0<\/span>is used. Hence, in our above example, we could write [latex]5 \\in\\{0,1,2,3,4,5,6,7\\}[\/latex], whereas [latex]8 \\notin\\{0,1,2,3,4,5,6,7\\}[\/latex].<\/span><\/p>\n

      \n

      Example 1<\/h3>\n

      Write <\/span>the first four\u00a0<\/span>months of a year as set M <\/span>using the List Method,\u00a0<\/span>identify one of <\/span>the months\u00a0<\/span>as <\/span>an element of the set<\/span>, and identify a month that is not an element of the set<\/span>.<\/span><\/p>\n

      Show Answer<\/span><\/p>\n
      \u00a0M= {January, February, March, April}
      \nFebruary [latex]\\in[\/latex] M
      \nJuly [latex]\\notin[\/latex] M\n<\/div>\n<\/div>\n<\/div>\n

      A set that has no elements i<\/span>s called an empty set and is denoted by the <\/span>symbol <\/span>\u00d8<\/span>.\u00a0<\/span><\/p>\n

      Find the intersection and union of sets<\/h2>\n

      Sets can be joined together using the intersection of sets or the union of sets.\u00a0<\/span><\/p>\n

      \n

      The intersection <\/span><\/span>of two sets A and B is the set of all elements that are common to both A and B<\/span> <\/span>and is denoted as <\/span>A\u00a0<\/span>\u2229\u00a0<\/span>B.<\/span><\/strong><\/p>\n

       <\/p>\n

      The union<\/span> of two sets A and B is the set of all elements in A or B<\/span> and is denoted as <\/span>A <\/span>\u222a\u00a0<\/span>B<\/span><\/strong><\/p>\n<\/div>\n

       <\/p>\n

      \n

      Example 2<\/h3>\n

      Find the intersection: {2, 3, 4, 5, 6, 7} <\/span>\u2229\u00a0<\/span>{1, 2, 5, 7, 8, 9}
      \n<\/span><\/p>\n

      Show Answer<\/span><\/p>\n
      {2, 5, 7}<\/span><\/p>\n

      Since elements 2, 5 and 7 were common to both sets, they are the elements in the intersection.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Example 3<\/h3>\n

      Find the union: <\/span>{2, 4, 6, 8} <\/span>\u222a\u00a0<\/span>{2, 3, 5, 7}<\/span><\/p>\n

      Show Answer<\/span><\/p>\n
      {2, 3, 4, 5, 6, 7, 8}<\/span><\/p>\n

      To create the union between the two sets, we will take all elements that are in either the first set or the second set. Notice, even though the 2 was in both sets, we only write it once in the union.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      Use interval notation to describe sets of numbers as intersections and unions<\/h2>\n

      When two inequalities are joined by the word and<\/i>, the solution of the compound inequality occurs when both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or<\/i>, the solution of the compound inequality occurs when either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.<\/p>\n

      In this module we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.<\/p>\n

      Venn diagrams use the concept of intersections and unions to show how much two or more things share in common. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently Cecilia has both of these qualities; therefore she is the intersection of the two.<\/p>\n

      \"Two<\/p>\n

      In mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers x<\/em> can be, and what would the interval look like?<\/p>\n

      In words, x<\/em> must be less than 6 and at the same time,<\/em> it must be greater than 2, much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Let’s look at a graph to see what numbers are possible with these constraints.<\/p>\n

      \"Number<\/p>\n

      The numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a bounded<\/em> inequality and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. x<\/em> must be less than 6 and greater than two\u2014the values for x<\/em> will fall between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:<\/p>\n

      \"Number<\/p>\n

      On the other hand, if you need to represent two things that\u00a0don’t share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements but are often considered in the same application, such as online shopping or banking.<\/p>\n

      \"Two<\/p>\n

      In mathematical terms, for example, [latex]x>6[\/latex]\u00a0or<\/em>\u00a0[latex]x<2[\/latex] is an inequality joined by the word “or”. Using interval notation, we can describe each of these inequalities separately:<\/em><\/em><\/p>\n

      [latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x<2[\/latex] is the same as\u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0or<\/em> real numbers greater than 6. Can you see why we need to write them as two separate intervals? Let’s look at a graph to get a clear picture of what is going on.<\/p>\n

      \"Number<\/p>\n

      When you place both of these inequalities on a graph, we can see that they share no numbers in common. This is what we call a union, as mentioned above. The interval notation associated with a union is a big U, so instead of writing or<\/em>, we join our intervals with a big U, like this:<\/p>\n

      [latex]\\left(-\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]<\/p>\n

       <\/p>\n

      \n

      Example 4<\/h3>\n

      Draw the graph of the compound inequality [latex]x\\gt3[\/latex] or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of x<\/em>-values that will satisfy it with an interval.<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      The graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on 3 and a blue arrow drawn to the right to contain all the numbers greater than 3.<\/p>\n

      \"Number<\/p>\n

      The graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than 4.<\/p>\n

      \"Number<\/p>\n

      What do you notice about the graph that combines these two inequalities?<\/p>\n

      \"Number<\/p>\n

      Since this compound inequality is an or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than 3 and less than or equal to 4 is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] or [latex]x\\le4[\/latex] is the set of all real numbers, and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex]<\/p>\n

      A final graph that can be drawn to represent the solution is shown below:<\/p>\n

      \"A<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      In the following video you will see two examples of how to express inequalities involving OR graphically and as an interval.
      \n