{
  "translations": {
    "en": {
      "ex1": {
        "type": "select",
        "description": "Lets see some examples to explain this. Suppose that $L_1 = \\{$fish, dog, carrot$\\}$ and that $L_2 = \\{$rot, g$\\}$.",
        "question": "What words in $L_1$ end with some string from $L_2$?",
        "answer": ["carrot", "dog"],
        "choices": ["fish", "dog", "carrot"]
      },
      "ex1b": {
        "type": "select",
        "description": "The right quotient $L_1 \\backslash L_2 = \\{x\\ |\\ xy \\in L_1$ for some $\\ y \\in L_2\\}$<br/><br/>$L_1 = \\{$fish, carrot, dog$\\}$, $L_2 = \\{$rot, g$\\}$.",
        "question": "What are the strings in $L_1 \\backslash L_2$?",
        "answer": ["car", "do"],
        "choices": ["fish", "dog", "do", "g", "rot", "carrot", "car"],
        "correctFeedback": ["'car' is what is left after cutting off 'rot' from 'carrot', and 'do' is left after cutting off 'g' from 'dog'."],
        "incorrectFeedback": ["You want the strings that you get by cutting off the part in $L_2$ that is at the end of a word in $L_1$. Do this for any strings of $L_1$ that end in either 'rot' or 'g'. There are two such words in $L_1$, and you want just what is left after the matching ending is removed."]
      },
      "ex2": {
        "type": "select",
        "description": "Lets try another example. Suppose that $L_1 = \\{$carrot, parrot, rot$\\}$ and that $L_2 = \\{$rot$\\}$.",
        "question": "What are the words in $L_1$ that end with words in $L_2$?",
        "answer": ["carrot", "parrot", "rot"],
        "choices": ["carrot", "parrot", "rot"]
      },
      "ex2b": {
        "type": "select",
        "description": "The right quotient $L_1 \\backslash L_2 = \\{x\\ |\\ xy \\in L_1$ for some $\\ y \\in L_2\\}$<br/><br/>$L_1 = \\{$carrot, parrot, rot$\\}$, $L_2 = \\{$rot$\\}$.",
        "question": "What is $L_1 \\backslash L_2$?",
        "answer": ["$\\lambda$", "car", "par"],
        "choices": ["carrot", "parrot", "rot", "$\\lambda$", "car", "par"],
        "incorrectFeedback": ["You want the strings that you get by cutting off 'rot' (the only string in $L_2$) from each string in $L_1$ that ends in 'rot' (which is all of them). If we cut off the whole string, what is left is $\\lambda$."]
      },
      "ex3": {
        "type": "select",
        "description": "Suppose that $L_1 = \\{$carrot$\\}$ and that $L_2 = \\{$t, ot$\\}$.",
        "question": "What is $L_1 \\backslash L_2$?",
        "answer": ["carr", "carro"],
        "choices": ["carrot", "car", "rot", "$\\lambda$", "carr", "carro"],
        "incorrectFeedback": ["This is just a little tricky, because it shows that multiple strings from $L_2$ can operate on a given string from $L_1$. So there are two strings in the answer for this example."]
      },
      "ex4": {
        "type": "select",
        "description": "Suppose that $L_1 = \\{$cab, dab$\\}$ and that $L_2 = \\{$b, ab$\\}$.",
        "question": "What are the words in $L_1 \\backslash L_2$?",
        "answer": ["c", "d", "ca", "da"],
        "choices": ["a", "b", "c", "d", "ca", "da", "cab", "dab"]
      },
      "lambda": {
        "type": "multiple",
        "description": "Remember that $L_1 \\backslash L_2 = \\{x\\ |\\ xy \\in L_1$ for some $\\ y \\in L_2\\}$. And note that $\\lambda$ is a perfectly good string, and could be a member of either $L_1$ or $L_2$.",
        "question": "If $\\lambda \\in L_2$, which of the following is a part of $L_1 \\backslash L_2$?",
        "answer": "$L_1$",
        "choices": ["$L_2$", "$L_1$", "$\\lambda$" ]
      },
      "complexex": {
        "type": "multiple",
        "description": "Here is our last example. Suppose that $L_1 = \\{a^*b^* \\cup b^*a^*\\}$ and that<br/>$L_2 = \\{b^n\\ |\\ n$ is even, $n > 0 \\}$.",
        "question": "What is $L_1 \\backslash L_2$?",
        "answer": "$\\{a^*b^*\\}$",
        "choices": ["$\\{a^*b^*\\}$","$\\{b^*a^*\\}$", "$\\{b^n\\ |\\ n$ is even, $n > 0 \\}$"],
        "correctFeedback": ["You should make sure that you really understand why before going on, or the rest of this will not make any sense to you. Look at the two parts that are union'ed in $L_1$, and consider how the right quotient operator affects each of these."]
      },
      "different": {
        "type": "multiple",
        "description": "Assume that we have a DFA called $M$ that accepts $L_1$. Our goal is to construct a DFA that accepts $L_1 \\backslash L_2$, and we will call this machine $M'$. It turns out that $M' = (Q, \\Sigma, \\delta, q_0, F')$ where $M = (Q, \\Sigma, \\delta, q_0, F)$.",
        "question": "What is the only difference between $M$ (the DFA for $L_1$) and $M'$ (the DFA for $L_1 \\backslash L_2$)?",
        "answer": "Different final states",
        "choices": ["They are identical", "Differnt transition function", "Different states", "Different final states"]
      },
      "q0": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_0$?",
        "answer": "No",
        "choices": ["Yes", "No"],
        "incorrectFeedback": ["It is not possible to get to a final state when starting at q0 without including an 'a' character."]
      },
      "q0final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_0$ be labeled as a final state?",
        "answer": "No",
        "choices": ["Yes", "No"],
        "correctFeedback": ["Because there is no walk from $q_0$ to a final state that is labeled with one or more $b$'s."]
      },
      "q1": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_1$?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "q1final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_1$ be a final state?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "q2": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_2$?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "q2final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_2$ be a final state?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "q3": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_3$?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "q3final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_3$ be a final state?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "q4": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_4$?",
        "answer": "No",
        "choices": ["Yes", "No"],
        "correctFeedback": ["Of course not. This was the trap state, so there is no path out to any final state in $M$."]
      },
      "q4final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_4$ be a final state?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "q5": {
        "type": "multiple",
        "description": "Let us test every state to see if it has a walk labled with strings $\\in L_2 = \\{b^m\\ |\\ m \\ge 1 \\}$ to any final state",
        "question": "Is there a walk to a final state labeled with one or more $b$'s for $q_5$?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "q5final": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state.",
        "question": "Should $q_5$ be a final state?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "lastex": {
        "type": "multiple",
        "description": "Let's do one last example. Suppose that $L_1 = \\{a^*baa^*\\}$ and $L_2 = \\{ab^*\\}$. What is $L_1 \\backslash L_2$?",
        "question": "Which DFA should we build first?",
        "answer": "The DFA for $L_1$",
        "choices": ["The DFA for $L_1$", "The DFA for $L_2$", "The DFA for $L_1 \\backslash L_2$"]
      },      
      "q0again": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state. $L_1 = \\{a^*baa^*\\}$ and $L_2 = \\{ab^*\\}$.",
        "question": "Let $L_0'$ to be the language that is represented by the DFA where $q_0$ is the start state. Can this machine accept anything in $L_2$? (If so, then $q_0$ should be a final state.)",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q1again": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state. $L_1 = \\{a^*baa^*\\}$ and $L_2 = \\{ab^*\\}$.",
        "question": "Let $L_1'$ to be the language that is represented by the DFA where $q_1$ is the start state. Can this machine accept anything in $L_2$? (If so, then $q_1$ should be a final state.)",
        "answer": "Yes",
        "choices": ["No", "Yes"],
        "correctFeedback": ["In this case, it is the string $a$ to $q_2$ (no $b$'s)."]
      },
      "q2again": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state. $L_1 = \\{a^*baa^*\\}$ and $L_2 = \\{ab^*\\}$.",
        "question": "Let $L_2'$ to be the language that is represented by the DFA where $q_2$ is the start state. Can this machine accept anything in $L_2$? (If so, then $q_2$ should be a final state.)",
        "answer": "Yes",
        "choices": ["No", "Yes"],
        "correctFeedback": ["Again, it is the string $a$ to $q_2$."]
      },
      "q3again": {
        "type": "multiple",
        "description": "The algorithm is: for each state $q_i$, if there is a walk labeled with strings $\\in L_2$ to a final state in $M$, we mark $q_i$ as final state. $L_1 = \\{a^*baa^*\\}$ and $L_2 = \\{ab^*\\}$.",
        "question": "Let $L_3'$ to be the language that is represented by the DFA where $q_3$ is the start state. Can this machine accept anything in $L_2$? (If so, then $q_3$ should be a final state.)",
        "answer": "No",
        "choices": ["No", "Yes"]
      }
    }
  }
}