{
  "translations": {
    "en": {
      "equivalent": {
        "type": "select",
        "description": "$R$ is an equivalence relation on set $S$ if it is reflexive, symmetric, and transitive.",
        "question": "Which of the following is an equivalence relation?",
        "answer": ["$=$"],
        "choices": ["$=$", "$<$", "$>$", "$\\leq$", "$\\geq$"]
      },
      "eqclass": {
        "type": "select",
        "description": "An equivalence relation can be used to partition a set into equivalence classes. If two elements $a$ and $b$ are equivalent to each other, we write $a \\equiv b$.",
        "question": "Why is $=$  an equivalence relation?",
        "answer": ["because it is reflective", "because it is symmetric", "because it is transitive"],
        "choices": ["because it is reflective", "because it is symmetric", "because it is transitive", "because it is antisymmetric", "because it is irreflexive"],
        "correctFeedback": ["$=$ also happens to be antisymmetric. But that is irrelevant to whether it is an equivalence relation."]
      },
      "mod": {
        "type": "multiple",
        "description": "For the set of integers, the modulus function defines a binary relation such that two numbers $x$ and $y$ are in the relation if and only if $x \\bmod m = y \\bmod m$",
        "question": "[T/F] For $m = 4$, the pair $\\langle 1, 5\\rangle$ belongs to the modulus relation.",
        "answer": "True",
        "choices": ["True", "False"],
        "incorrectFeedback": ["Is $1 \\bmod 4$ different from $5 \\bmod 4$?"]
      },
      "partorder": {
        "type": "select",
        "description": "A binary relation is called a partial order if it is antisymmetric and transitive.",
        "question": "Which of the following relations defines a partial order?",
        "answer": ["$<$", "$\\leq$", "$=$"],
        "choices": ["$<$", "$\\leq$", "$=$"],
        "correctFeedback": ["It might seem odd to think that $=$ is a partial order. But it is in fact vacuously antisymmetric (there is never a case where $a=b$ and $b=a$ unless they are in fact the same thing) and vacuously transitive (again, $a=b$ and $b=c$ can only be true if $a=c$ because they are all the same)."]
      }
    }
  }
}