{
"translations": {
"en": {
"add": {
"type": "select",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of adding two positive even integers?",
"answer": ["Another positive even integer", "A number $\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"addclosed": {
"type": "multiple",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "So, is $L$ closed under addition?",
"answer": "Yes",
"choices": ["Yes", "No"]
},
"multiply": {
"type": "select",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of multiplying two positive even integers?",
"answer": ["Another positive even integer", "A number $\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"multclosed": {
"type": "multiple",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "So, is $L$ closed under multiplication?",
"answer": "Yes",
"choices": ["Yes", "No"]
},
"subtract": {
"type": "select",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of subtracting a positive even integer from another? A couple of examples are $6 - 4$ and $4 - 6$.",
"answer": ["Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"subnotclosed": {
"type": "multiple",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "So, is $L$ closed under subtraction?",
"answer": "No",
"choices": ["Yes", "No"],
"correctFeedback": ["To be closed, it has to be true for any two members of the set."]
},
"div": {
"type": "select",
"description": "Definition: A set is closed over a (binary) operation if, whenever that operation is applied to any two members of the set, the result is a member of the set.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of dividing one positive even integer by another?",
"answer": ["Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"generalize": {
"type": "select",
"description": "So far, we have been looking at binary operators that we generally think of as written in infix notation. A more general way to write this is like a function call, for example we might write addition as addition(x, y). And, we don't need to stick to binary operators. We can do unary operators, for example. Such as an operator to make a value negative: negative(x).",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of negative(x) for $x \\in L$?",
"answer": ["A number $\\not\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"increment": {
"type": "select",
"description": "Consider the operator ++(x), which increments $x$ by one.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of ++(x) for $x \\in L$?",
"answer": ["A number $\\not\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"double": {
"type": "select",
"description": "Consider the operator double(x), which doubles the value of $x$.",
"question": "Suppose $L = \\{x\\ |\\ x$ is a positive even integer$\\}$. What is the result of double(x) for $x \\in L$?",
"answer": ["Another positive even integer", "A number $\\in L$"],
"choices": ["Another positive even integer", "A number $\\in L$", "A number $\\not\\in L$", "Sometimes a number $\\in L$, and sometimes a number $\\not\\in L$"]
},
"strings": {
"type": "select",
"description": "Of course, we can also talk about sets of strings. Suppose $L = \\{x\\ |\\ x$ is a string containing any even number of a's.$\\}$.",
"question": "For strings $x$ and $y$ in $L$, what is the result of $x \\cup y$?",
"answer": ["A string", "A string $\\in L$"],
"choices": ["A string", "A string $\\in L$", "A string $\\not\\in L$", "Sometimes a string $\\in L$, and sometimes a string $\\not\\in L$"]
},
"addstrings": {
"type": "select",
"description": "Of course, we can also talk about sets of strings. Suppose $L = \\{x\\ |\\ x$ is a string containing any even number of a's.$\\}$.",
"question": "Consider binary operator addition that adds two numbers. What is addition(x, y) for $x$, $y$ in $L$?",
"answer": ["This makes no sense"],
"choices": ["This makes no sense", "A string $\\in L$", "A string $\\not\\in L$", "Sometimes a string $\\in L$, and sometimes a string $\\not\\in L$"],
"correctFeedback": ["$x$ and $y$ are strings, and addition is defined on numbers. So, $L$ is not closed on addition."]
},
"regular": {
"type": "multiple",
"description": "In the rest of this module, we are going to talk about whether certain operators are closed or not for regular languages.",
"question": "What is a regular language?",
"answer": "A set of strings",
"choices": ["A set of strings"],
"correctFeedback": ["Like any language, a regular language is just a set of string."]
}
}
}
}