{
    "translations": {
      "en": {
        "tuple": {
          "type": "select",
          "description": "A relation $R$ over set $S$ means a set of :term:`tuples` of elements from set $S$. A tuple is just some elements of the set in $\\langle\\rangle$ brakets.",
          "question": "If $S$ is $\\{a, b, c\\}$, which of these are tuples?",
          "answer": ["$\\langle a, b, c\\rangle$", "$\\langle a, c\\rangle$", "$\\langle a \\rangle$"],
          "choices": ["$\\langle a, b, c\\rangle$", "$\\langle a, c\\rangle$", "$\\langle a \\rangle$", "$\\{a, c\\}$", "$\\langle a, d\\rangle$"]
        },
        "binary": {
          "type": "select",
          "description": "While a relation might work on an number of items from set $S$ (just like a function might require any number of parameters), nearly always we are going to consider :term:`binary relations`. In this case, the relation is a set of ordered pairs, where the elements in the ordered pair are elements of $S$. We often use relations such as 'plus' ($+$) or the less-than operator ($<$) on the natural numbers.",
          "question": "The 'less than' operator ($<$) includes which of the following ordered pairs?",
          "answer": ["$\\langle1, 3\\rangle$", "$\\langle2, 23\\rangle$"],
          "choices": ["$\\langle1, 3\\rangle$", "$\\langle2, 23\\rangle$", "$\\langle 3, 2\\rangle$", "$\\langle2, 2\\rangle$"],
          "correctFeedback": ["An ordered pair is in the $<$ relation if and only if the first member of the pair is less than the second."],
          "incorrectFeedback": ["An ordered pair is in the $<$ relation if and only if the first member of the pair is less than the second."]
        },
        "infix": {
          "type": "multiple",
          "description": "Rather than writing binary relationship in terms of ordered pairs, we can also use infix notation for such relations.",
          "question": "What is the infix notation for the pair $\\langle 4, 5\\rangle$ in the relation of $<$?",
          "answer": "4 $<$ 5",
          "choices": ["4 $<$ 5", "5 $<$ 4", "4 $=$ 5", "4 $\\le$ 5", "4 $\\ge$ 5"]
        },
        "relation": {
          "type": "select",
          "description": "For the rest of this discussion, we assume that our relations are always binary, and so the relation is a set of ordered pairs. And remember that a relation $R$ on set $S$ is simply some subset of the ordered pairs possible on elemnts of $S$. This might mean none of them (the empty set of such ordered pairs) all of them, or anything in between.",
          "question": "For set $S = \\{1, 2, 3\\}$, which of these is the set of ordered pairs making up the $<$ relation?",
          "answer": ["$\\langle 1, 2\\rangle$", "$\\langle 1, 3\\rangle$", "$\\langle 2, 3\\rangle$"],
          "choices": ["$\\langle 1, 1\\rangle$", "$\\langle 1, 2\\rangle$", "$\\langle 1, 3\\rangle$", "$\\langle 2, 1\\rangle$", "$\\langle 2, 2\\rangle$", "$\\langle 2, 3\\rangle$", "$\\langle 3, 1\\rangle$", "$\\langle 3, 2\\rangle$", "$\\langle 3, 3\\rangle$"]
        },
        "reflex": {
          "type": "multiple",
          "description": "$R$ is reflexive if $aRa$ for all $a \\in S$.",
          "question": "[T/F] The relation 'less than or equal to' ($\\leq$) is reflexive.",
          "answer": "True",
          "choices": ["True", "False"],
          "correctFeedback": ["For all $a \\in S$, $a\\leq a$ is true."],
          "incorrectFeedback": ["For all $a \\in S$, $a\\leq a$ is true."]
        },
        "irreflexleq": {
          "type": "multiple",
          "description": "$R$ is irreflexive if $aRa$ is not true for all $a \\in S$.",
          "question": "[T/F] The relation 'less than or equal to' ($\\leq$) is irreflexive.",
          "answer": "False",
          "choices": ["True", "False"],
          "correctFeedback": ["For all $a \\in S$, $a\\leq a$ is true. In other words, it is <b>not true</b> for all $a \\in S$ that $a\\leq a$ is false, which is what the irreflexive property demands."],
          "incorrectFeedback": ["For all $a \\in S$, $a\\leq a$ is true."]
        },
        "irreflexle": {
          "type": "multiple",
          "description": "$R$ is irreflexive if $aRa$ is not true for all $a \\in S$.",
          "question": "[T/F] The relation 'less than' ($<$) is irreflexive.",
          "answer": "True",
          "choices": ["True", "False"],
          "correctFeedback": ["For all $a \\in S$, $a < a$ is false."],
          "incorrectFeedback": ["For all $a \\in S$, $a < a$ is false."]
        },
        "symmetric": {
          "type": "select",
          "description": "$R$ is symmetric if whenever $aRb$, then $bRa$ for all $a, b \\in S$.",
          "question": "Which of the following is a symmetric relation?",
          "answer": ["$=$"],
          "choices": ["$<$", "$>$", "$=$"]
        },
        "antisymmetric": {
          "type": "select",
          "description": "$R$ is antisymmetric if whenever $aRb$ and $bRa$, then $a = b$, for all $a, b \\in S$",
          "question": "Which of the following relations are antisymmetric?",
          "answer": ["$>$", "$=$", "$\\leq$"],
          "choices": ["$>$", "$=$", "$\\leq$"]
        },
        "transitive": {
          "type": "select",
          "description": "$R$ is transitive if whenever $aRb$ and $bRc$, then $aRc$, for all $a, b, c \\in S$.",
          "question": "Which of the followings are transitive relations?",
          "answer": ["$<$", ">", "="],
          "choices": ["$<$", ">", "="],
          "correctFeedback": ["For $=$, this is vacuously true."]
        }
      }
    }
  }