{
  "translations": {
    "en": {
      "braces": {
        "type": "multiple",
        "description": "We always use $\\{\\}$ braces when we describe a set.",
        "question": "Which of the following is a valid description of a set?",
        "answer": "$\\{1, 2, 3\\}$",
        "choices": [
          "$1, 2, 3$",
          "$\\{1, 2, 3\\}$"
        ]
      },
      "former": {
        "type": "multiple",
        "description": "One way to describe a set is to write out all of it's elements. That is a bit hard to do if the set has an infinite number of elements. Another way to describe a set is to use a :term:`set former` like this: $\\{x\\ |\\ x$ is a positive integer$\\}$.",
        "question": "Choose which best describes this set.",
        "answer": "all positive integers",
        "choices": [
          "all integers",
          "all negative integers",
          "all positive integers",
          "$\\{1, 2, 3\\}$"
        ]
      },
      "members": {
        "type": "select",
        "description": "$x \\in P$ means that $x$ is a member of set $P$. $x \\notin P$ means that  $x$ is not a member of set $P$",
        "question": "Let $P = \\{a, b, c\\}$, which of the following are $\\notin P$?",
        "answer": ["$\\{a\\}$", "$\\{a, b, c\\}$"],
        "choices": ["$a$", "$\\{a\\}$", "$b$", "$\\{a, b, c\\}$"]
      },
      "null": {
        "type": "select",
        "description": "$\\emptyset$ is the symbol for the null set or the empty set. Alternatively, we might write the empty set as $\\{\\}$.",
        "question": "Which of the following is $\\in \\emptyset$?",
        "answer": ["$\\emptyset$ has no elements because it is the null set"],
        "choices": [
          "$a$",
          "1",
          "$\\{a, b\\}$",
          "$\\{\\}$",
          "$\\emptyset$ has no elements because it is the null set"
        ],
        "correctFeedback": ["Note that $\\{\\}$ is not in $\\emptyset$ because $\\emptyset$ has no members, not even the empty set."]
      },
      "size": {
        "type": "multiple",
        "description": "$|P|$ means size or cardinality of set $P$.",
        "question": "What does it mean to say that $|P| = 3$?",
        "answer": "P has 3 elements",
        "choices": [
          "P has 3 elements",
          "P is 3",
          "P has an element with value 3"
        ]
      },
      "size2": {
        "type": "multiple",
        "description": "$|P|$ means size or cardinality of set $P$.",
        "question": "What is $|\\{3, 4, 4, 5\\}|$?",
        "answer": 3,
        "choices": [1, 2, 3, 4, 5, "none of these"],
        "correctFeedback": ["A set has no concept of duplicate elements."]
      },
      "subset": {
        "type": "select",
        "description": "The notation $P \\subseteq Q$ or $Q \\supseteq P$ both mean that set $P$ is included in set $Q$. That is, set $P$ is a subset of set $Q$ ($\\subseteq$), or set $Q$ is a superset of the set $P$ ($\\supseteq$).",
        "question": "Let $P = \\{x\\ |\\ x\\ \\mbox{is a positive integer}\\}$. Which of the following is a subset of $P$?",
        "answer": [
          "$\\{1, 2, 3\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a positive even number}\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a prime number}\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a positive integer}\\}$"],
        "choices": [
          "$\\{1, 2, 3\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a positive even number}\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a real number}\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a prime number}\\}$",
          "$\\{x\\ |\\ x\\ \\mbox{is a positive integer}\\}$"
        ],
        "correctFeedback": ["When using symbol $\\subseteq$, a subset of $P$ can be equal to $P$. On the other hand, the symbol $\\subset$ refers to a <b>proper</b> subset of $P$, meaning that the subset cannot equal $P$."]
      },
      "union": {
        "type": "multiple",
        "description": "$P \\cup Q$ means set union: All elements appearing in $P$ OR $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 5, 6\\}$. What is $P \\cup Q$?",
        "answer": "$\\{1, 2, 3, 4, 5, 6\\}$",
        "choices": [
          "$\\{1, 2, 3, 4, 5, 6\\}$",
          "$\\{1, 3, 5\\}$",
          "$\\{2, 4, 5, 6\\}$",
          "$\\{5\\}$"
        ]
      },
      "emptyunion": {
        "type": "select",
        "description": "$P \\cup Q$ means set union: All elements appearing in $P$ OR $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 5, 6\\}$. What is $Q \\cup P$?",
        "answer": ["$\\{1, 2, 3, 4, 5, 6\\}$", "$\\{1, 2, 3, 4, 5, 5, 6\\}$"],
        "choices": [
          "$\\{\\}$",
          "$\\{1, 2, 3, 4, 5, 6\\}$",
          "$\\{1, 2, 3, 4, 5, 5, 6\\}$"
        ],
        "correctFeedback": ["Remember that sets have no concept of duplicate elements, so it doesn't matter how many times you list them, it's the same set."]
      },
      "commute": {
        "type": "multiple",
        "description": "$P \\cup Q$ means set union: All elements appearing in $P$ OR $Q$",
        "question": "[T/F] $P \\cup Q = Q \\cup P$",
        "answer": "True",
        "choices": ["True", "False"],
        "correctFeedback": ["Union is commutative."]
      },
      "intersect": {
        "type": "multiple",
        "description": "$P \\cap Q$ means set intersection: All elements appearing in $P$ AND $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 5, 6\\}$. What is $P \\cap Q$?",
        "answer": "$\\{5\\}$",
        "choices": [
          "$\\{1, 2, 3, 4, 5, 6\\}$",
          "$\\{1, 3, 5\\}$",
          "$\\{2, 4, 5, 6\\}$",
          "$\\{5\\}$",
          "$5$",
          "$\\{\\}$"
        ]
      },
      "emptyintersect": {
        "type": "multiple",
        "description": "$P \\cap Q$ means set intersection: All elements appearing in $P$ AND $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 6\\}$. What is $P \\cap Q$",
        "answer": "$\\{\\}$",
        "choices": [
          "$\\{1, 2, 3, 4, 5, 6\\}$",
          "$\\{1, 3, 5\\}$",
          "$\\{2, 4, 5, 6\\}$",
          "$\\{\\}$"
        ]
      },
      "commuteI": {
        "type": "multiple",
        "description": "$P \\cap Q$ means set intersection: All elements appearing in $P$ AND $Q$",
        "question": "[T/F] $P \\cap Q = Q \\cap P$",
        "answer": "True",
        "choices": ["True", "False"],
        "correctFeedback": ["Intersection is also commutative."]
      },
      "setdiff": {
        "type": "multiple",
        "description": "$P - Q$ means set difference: all elements of set $P$ NOT in set $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 5, 6\\}$. What is $P - Q$?",
        "answer": "$\\{1, 3\\}$",
        "choices": [
          "$\\{1, 3\\}$",
          "$\\{1, 3, 5\\}$",
          "$\\{2, 3, 5, 6\\}$",
          "$\\{2, 4, 6\\}$",
          "$\\{1, 2, 3, 4, 6\\}$"
        ],
        "correctFeedback": ["Extra elements in $Q$ (like 2, 4, and 6 in this example) do not matter. Only the 5 matters here, it is removed from $P$ in the answer."]
      },
      "setdiff2": {
        "type": "multiple",
        "description": "$P - Q$ means set difference: all elements of set $P$ NOT in set $Q$",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4, 5, 6\\}$. What is $Q - P?$",
        "answer": "$\\{2, 4, 6\\}$",
        "choices": [
          "$\\{1, 3\\}$",
          "$\\{1, 3, 5\\}$",
          "$\\{2, 3, 5, 6\\}$",
          "$\\{2, 4, 6\\}$",
          "$\\{1, 2, 3, 4, 6\\}$"
        ]
      },
      "setdiffcomm": {
        "type": "multiple",
        "description": "$P - Q$ means set difference: all elements of set $P$ NOT in set $Q$",
        "question": "[T/F] $P - Q = Q - P$",
        "answer": "False",
        "choices": [
          "True",
          "False"
        ],
        "correctFeedback": ["Set difference is not commutative."]
      },
      "product": {
        "type": "select",
        "description": "$P \\times Q$ means set (Cartesian) Product. This operation yields a set of ordered pairs.",
        "question": "Let $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4\\}$ What is $P \\times Q$? Note that if there are 3 elements in $P$ and 2 elements in $Q$, then there must be $2 \\times 3 = 6$ elements in $P \\times Q$.",
        "answer": ["$\\{(1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5, 4)\\}$"],
        "choices": [
          "$\\{(1, 2), (1, 4)\\}$",
          "$\\{(1, 2), (1, 4), (3, 2)\\}$",
          "$\\{(1, 4), (3, 2), (5, 2), (5, 4)\\}$",
          "$\\{(1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5, 4)\\}$"
        ]
      },
      "productsize": {
        "type": "multiple",
        "description": "$P \\times Q$ means set (Cartesian) Product. This operation yields a set of ordered pairs.",
        "question": "For $P = \\{1, 3, 5\\}$, and $Q = \\{2, 4\\}$, what is $|P \\times Q|$?",
        "answer": "6",
        "choices": ["1", "2", "3", "4", "5", "6", "7"]
      },
      "powerset": {
        "type": "multiple",
        "description": "The powerset of set $S$ (written $2^S$ ) is the set of all possible subsets for $S$",
        "question": "For set $S = \\{a, b, c\\}$, what is $2^S$?",
        "answer": "$\\{\\emptyset, \\{a\\}, \\{b\\}, \\{c\\}, \\{a, b\\}, \\{a, c\\}, \\{b, c\\}, \\{a, b, c\\}\\}$",
        "choices": [
          "$\\{\\emptyset\\}$",
          "$\\{\\emptyset, \\{a\\}, \\{b\\}, \\{c\\}, \\{a, b\\}, \\{a, c\\}, \\{b, c\\}, \\{a, b, c\\}\\}$",
          "$\\{\\emptyset, \\{a\\}, \\{b\\}, \\{c\\}\\}$",
          "$\\{\\emptyset, \\{a\\}, \\{b\\}, \\{c\\}, \\{a, b\\}, \\{a, c\\}\\}$"
        ]
      },
      "powersize": {
        "type": "multiple",
        "description": "The powerset of set $S$ (written $2^S$ ) is the set of all possible subsets for $S$",
        "question": "For set $S = \\{a, b, c\\}$, what is $|2^S|$?",
        "answer": "8",
        "choices": [
          "3",
          "9",
          "8",
          "10"
        ],
        "correctFeedback": ["If $|S| = n$, then $|2^S| = 2^n$. Which helps to explain why this notation is used."]
      }
    }
  }
}