{
"translations": {
"en": {
"alpha": {
"type": "select",
"description": "Let's start with an :term:`alphabet`. We normally represent an alphabet with the symbol $\\Sigma$ (Sigma). Alphabet $\\Sigma$ is just a set of symbols. The symbols we typically use are letters or digits, but an alphabet could include any symbols.",
"question": "Which of the following is an alphabet?",
"answer": ["$\\{a, b, c\\}$", "$\\{0, 1\\}$", "$\\{?, a, x\\}$"],
"choices": ["$\\{a, b, c\\}$", "$\\{0, 1\\}$", "$A$", "$\\{?, a, x\\}$"],
"correctFeedback": ["Any set of symbols is an alphabet."],
"incorrectFeedback": ["Make sure that you select all of the alphabets, and that everything you select is a set."]
},
"char":{
"type": "multiple",
"description": "Let's start with an :term:`alphabet`. We normally represent an alphabet with the symbol $\\Sigma$ (Sigma). Alphabet $\\Sigma$ is just a set of symbols. The symbols we typically use are letters or digits, but an alphabet could include any symbols.",
"question": "[T/F] $A$ is not an alphabet because it is just a character",
"answer": "True",
"choices": ["True", "False"]
},
"strings": {
"type": "select",
"description": "Given some alphabet set $\\Sigma$, a string is any finite sequence of symbols taken from $\\Sigma$.",
"question": "Suppose we have $\\Sigma$ = $\\{a, b\\}$. Which of the following are valid strings on $\\Sigma$?",
"answer": ["$babababbaba$", "$a$"],
"choices": ["$babababbaba$", "$a$", "$\\{aaa, bbb\\}$", "$bacbabcba$"],
"incorrectFeedback": ["Strings are not sets, and a valid string can only have characters from the alphabet."]
},
"stringnotset": {
"type": "multiple",
"description": "Given some alphabet set $\\Sigma$, a string is any finite sequence of symbols taken from $\\Sigma$.",
"question": "Suppose we have $\\Sigma = \\{a, b\\}$. Why is $\\{aaa, bbb\\}$ not a valid string?",
"answer": "Because $\\{aaa, bbb\\}$ is a set, not a string",
"choices": ["Because it has characters not in the alphabet", "Because $\\{aaa, bbb\\}$ is a set, not a string"]
},
"badchar": {
"type": "multiple",
"description": "Given some alphabet set $\\Sigma$, a string is any finite sequence of symbols taken from $\\Sigma$.",
"question": "Suppose we have $\\Sigma = \\{a, b\\}$. [T/F] $bacbabcba$ is a vaild string in $\\Sigma$",
"answer": "False",
"choices": ["True", "False"],
"correctFeedback": ["It contains characters not in the alphabet."]
},
"examplelang": {
"type": "select",
"description": "A language is simply a subset of the strings that can be defined over $\\Sigma$. We often use the variable $L$ to indicate a language.",
"question": "Which of the following are languages from their respective $\\Sigma$?",
"answer": [
"$\\Sigma = \\{0,1,2,3,4,5,6,7,8, 9\\}$, $L = \\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...\\}$",
"$\\Sigma = \\{a,b,c\\}$, $L=\\{ab,ac,cabb\\}$",
"$\\Sigma = \\{0,1,2,3,4,5,6,7,8, 9\\}$, $L = \\{set\\ of\\ all\\ positive\\ even\\ numbers\\}$"
],
"choices": [
"$\\Sigma = \\{0,1,2,3,4,5,6,7,8, 9\\}$, $L = \\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...\\}$",
"$\\Sigma = \\{a,b,c\\}$, $L=\\{ab,ac,cabb\\}$",
"$\\Sigma = \\{0,1,2,3,4,5,6,7,8, 9\\}$, $L = \\{set\\ of\\ all\\ positive\\ even\\ numbers\\}$"
],
"correctFeedback": ["Notice the different ways that we describe the strings that are in the language."]
},
"notset": {
"type": "multiple",
"description": "A language is simply a subset of the strings that can be defined over $\\Sigma$. We often use the variable $L$ to indicate a language.",
"question": "For $\\Sigma$=$\\{a,b,c\\}$, is $L = ab$ a language?",
"answer": "No",
"choices": ["Yes", "No"],
"correctFeedback": ["$ab$ is not a set."],
"incorrectFeedback": ["Careful! A language is a set of strings, but $ab$ is not a set."]
},
"isset": {
"type": "multiple",
"description": "A language is simply a subset of the strings that can be defined over $\\Sigma$.",
"question": "For $\\Sigma$=$\\{a,b,c\\}$, is $L = \\{ab\\}$ a language?",
"answer": "Yes",
"choices": ["Yes", "No"],
"correctFeedback": ["$\\{ab\\}$ is a set (with only one string)."],
"incorrectFeedback": ["$\\{ab\\}$ is a set of (one) string, and any set of strings is a language."]
},
"infinite": {
"type": "multiple",
"description": "For $\\Sigma$={$0, 1, 2, 3, 4, 5, 6, 7, 8, 9\\}$, we can define $L=\\{all\\ even\\ numbers\\}$. This is an example of a descriptive set former.",
"question": "How many strings are in this language?",
"answer": "An infinite number",
"choices": ["A finite number", "An infinite number"],
"correctFeedback": ["So, we cannot describe this language by writing out all of its strings."]
},
"empty": {
"type": "multiple",
"description": "A language is simply a subset of the strings that can be defined over $\\Sigma$.",
"question": "For $\\Sigma$=$\\{a,b,c\\}$, is $L$ = $\\{\\}$ a language?",
"answer": "Yes",
"choices": ["Yes", "No"]
},
"emptysymbol": {
"type": "multiple",
"description": "Right, since a language is just a set of strings, the empty set is a perfectly good language.",
"question": "Oh, and remember that we have a special way to write “empty” set. Which is it?",
"answer": "$\\{\\}$",
"choices": ["$\\Sigma$", "$E$", "$\\{\\}$"]
},
"stringname": {
"type": "multiple",
"description": "A language can have any symbols in its alphabet. But to get our ideas across, we nearly always use languages with simple alphabets of only a few characters. We typically use either digits or else letters near the start of the English alphabet. So, we nearly always use only $a, b, c, d, 0, 1$ in our alphabets. For string names (that is, some variable that is meant to refer to a string), we usually use letters near the end of the English alphabet: $u, v, w, x, y, z$",
"question": "So, when we talk about $b$, it probably refers to",
"answer": "The character ‘$b$’.",
"choices": ["The character ‘$b$’.", "Some string named $b$."]
},
"charname": {
"type": "multiple",
"description": "A language can have any symbols in its alphabet. But to get our ideas across, we nearly always use languages with simple alphabets of only a few characters. We typically use either digits or else letters near the start of the English alphabet. So, we nearly always use only $a, b, c, d, 0, 1$ in our alphabets. For string names (that is, some variable that is meant to refer to a string), we usually use letters near the end of the English alphabet: $u, v, w, x, y, z$",
"question": "And if we talk about $w$, it probably refers to ",
"answer": "Some string given the name $w$",
"choices": ["The character ‘$w$’", "Some string given the name $w$"]
},
"size": {
"type": "multiple",
"description": "We are going to use a few basic string operations a lot. The size of a string is the number of symbols in the string. We use the operator $|\\ |$ to donate the size of the string.",
"question": "Let $w=abbc$, $v=ab$, What is $|w|$ + $|v|$",
"answer": "6",
"choices": ["3", "4", "5", "6", "7", "8"],
"correctFeedback": ["$|w| = 4$, and $|v| = 2$. So, $4 + 2 = 6$."]
},
"concat": {
"type": "multiple",
"description": "Another important operation is string concatenation. The concatenation of two strings $u$ and $v$ is just the string that contains the symbols of $u$ followed by the symbols of $v$. We use $\\circ$ to represent concatenation. Example: Let $w = a_1 a_2 ... a_n$, and $v = b_1 b_2 ... b_m$ Then, $w \\circ v = a_1 a_2 ... a_n b_1 b_2 ... b_m$",
"question": "Let $w = abc$ and $v = 012$, What is $w \\circ v$?",
"answer": "$abc012$",
"choices": ["$abc012$", "$012abc$", "$012$", "$abc$"]
},
"nocirc": {
"type": "multiple",
"description": "In practice, we use concatenation so much that we hardly ever want to be so formal as to show the $\\circ$ operator. When we mean $w \\circ v$, we will normally simply write $wv$ instead.",
"question": "Let $w = abc$, $v = 012$, What is $wv$?",
"answer": "$abc012$",
"choices": ["$abc012$", "$012abc$", "$abc$", "$012$"]
},
"emptystring": {
"type": "multiple",
"description": "Since a language is just a subset of all possible strings made from the letters in $\\Sigma$, then the empty set is a language: $L$ = {}.",
"question": "But, is this a language? $\\{\\lambda\\}$",
"answer": "Yes",
"choices": ["Yes", "No"],
"correctFeedback": ["Right! Its the set that contains the empty string."],
"incorrectFeedback": ["What's your problem? The empty string is a perfectly good string, and a set that contains one string is a perfectly good set."]
},
"selfconcat": {
"type": "select",
"description": "Of course, we can also concatenate a string with itself. We talk about this often enough that its worthwhile to have notation for it. Suppose we have a string $v$. The notation $v^3$ means concatenate $v$ with itself 3 times. Concatenating a string with itself zero times yields the empty string $\\lambda$.",
"question": "let $v = ab$, what is $v^3$ ?",
"answer": ["$vvv$", "$v \\circ v \\circ v$", "$ababab$"],
"choices": ["$vvv$", "$v \\circ v \\circ v$", "$ababab$"],
"incorrectFeedback": ["Check all that apply."]
},
"identity": {
"type": "multiple",
"description": "Concatenating string $v$ with $\\lambda$ gives $v$. This means that $\\lambda$ is the concatenation identity operator. For example, $ab \\circ \\lambda = \\lambda \\circ ab = ab$",
"question": "What is $\\lambda \\circ acb \\circ \\lambda$?",
"answer": "$acb$",
"choices": ["$abc$","$acb$", "$bca$", "$bac$", "$cba$"]
},
"reverse": {
"type": "multiple",
"description": "Another useful operation is to reverse a string. Reversing a string means to write its symbols in reverse order. $w^R$ means to take the reverse of $w$.",
"question": "If $w = abbc$, then what is $w^R$?",
"answer": "$cbba$",
"choices": ["$abbc$", "$bcab$", "$abcb$", "$cbab$","$cbba$"]
},
"revlength": {
"type": "multiple",
"description": "Another useful operation is to reverse a string. Reversing a string means to write its symbols in reverse order. $w^R$ means to take the reverse of $w$.",
"question": "let $w=abbc$, and $v=ab$. What is $|vv^Rw|$?",
"answer": "$8$",
"choices": ["$8$", "$16$", "$24$", "$32$", "$40$"]
},
"closure": {
"type": "select",
"description": "One of the most important operations that we use is the * operator, also called the closure operator. “*” of something means concatenating that something zero or more times. For example, if we have $\\Sigma$ = {a,b}, $\\Sigma$* means the set of strings obtained by concatenating 0 or more symbols from $\\Sigma$. ",
"question": "For $\\Sigma = {a,b}$, which of the following are strings in $\\Sigma^∗$?",
"answer": ["$\\lambda$", "$a$", "$b$", "$aa$", "$abbabb$"],
"choices": ["$\\lambda$", "$a$", "$b$", "$aa$", "$abbabb$", "$aaabababcc$"],
"incorrectFeedback": ["Check all that apply."]
},
"closurex": {
"type": "select",
"description": "One of the most important operations that we use is the * operator, also called the closure operator. “*” of something means concatenating that something zero or more times. For example, if we have $\\Sigma$ = {a,b}, $\\Sigma$* means the set of strings obtained by concatenating 0 or more symbols from $\\Sigma$. ",
"question": "Which are examples of $w^*$ where $w = abc$?",
"answer": ["$\\lambda$", "$abc$", "$abcabc$"],
"choices": ["$\\lambda$", "$aabbcc$", "$abc$", "$abcabc$"],
"incorrectFeedback": ["Check all that apply."]
},
"plus": {
"type": "select",
"description": "Slightly different from the * operator, sometimes we want to have at least one copy of the thing that we are concatenating. Then we use +. One way to think about this: $\\Sigma^+ = \\Sigma^∗ − {\\lambda}$.",
"question": "Let $\\Sigma = \\{a, b\\}$, Which of the following are strings in $\\Sigma^+$?",
"answer": ["$a$", "$b$", "$aa$", "$abbabb$"],
"choices": [
"$\\lambda$",
"$a$",
"$b$",
"$aa$",
"$abbabb$",
"$aaabababcc$"
],
"incorrectFeedback": ["Check all that apply."]
},
"union": {
"type": "select",
"description": "Since languages are sets, all set operations can be applied to them.",
"question": " Let $\\Sigma$={$a$,$b$,$c$}, $L_1$={$ab$,$bc$,$aba$}, and $L_2$={$c$,$bc$,$bcc$}. Which strings are included in $L_1 \\cup L_2$ ?",
"answer": ["ab" ,"bc","aba","c","bcc"],
"choices": ["ab", "bc","a", "bca", "aba","c", "bcc","cb"],
"correctFeedback": ["$\\cup$ is the symbol for set union."],
"incorrectFeedback": ["$\\cup$ is the symbol for set union."]
},
"intersect": {
"type": "multiple",
"description": "Since languages are sets, all set operations can be applied to them.",
"question": "Let $\\Sigma=\\{a,b,c\\}$, $L_1=\\{ab,bc,aba\\}$, and $L_2=\\{c,bc,bcc\\}$. Which strings are in $L_1 \\cap L_2$ ?",
"answer": "$bc$",
"choices": ["$aba$", "$ab$", "$c$","$bc$", "$bcc$"],
"correctFeedback": ["$\\cap$ is the symbol for set intersection."],
"incorrectFeedback": ["$\\cap$ is the symbol for set intersection."]
},
"comp": {
"type": "select",
"description": "The complement of $L$ (written $\\bar{L}$) is the set of all strings in $\\Sigma^*$ except the strings in $L$. In other words, $\\bar{L}$ = $\\Sigma^* - L$",
"question": "Let $\\Sigma=\\{a,b,c\\}$, $L_1=\\{ab,bc,aba\\}$, and $L_2=\\{c,bc,bcc\\}$. What is $\\overline{L_1∩L_2}$?",
"answer": "$\\Sigma^* - \\{bc\\}$",
"choices": ["$\\{bc\\}$", "$\\Sigma^* - \\{bc\\}$", " $\\Sigma^* \\cup \\{bc\\}$", "$\\Sigma - \\{bc\\}$"]
},
"concatsets": {
"type": "multiple",
"description": "Sometimes we want to concatenate two sets (as opposed to concatenating two strings). Concatenation of sets $L_1$ and $L_2$ is written $L_1 \\circ L_2$ or $L_1 L_2$. The result is the set created by concatenating each string in $L_1$ with each string in $L_2$.",
"question": "Let $L_1=\\{ab,bc,aba\\}$ and $L_2=\\{c,bc,bcc\\}$. Which of the following is $L_1 \\circ L_2$?",
"answer": "$\\{abc,abbc,abbcc,bcc,bcbc,bcbcc,abac,ababc,ababcc\\}$",
"choices": [ "$\\{abc,abbc,abbcc,bcc,bcbc,bcbcc,abac,ababc,abab\\}$","$\\{abc,abbc,abbcc,bcc,bcbc,bcbcc,abac,ababc,ababcc\\}$", "$\\{abc,abbc,abbcc,bcc,bbcc,bcbcc,abac,abab,ababcc\\}$"]
},
"selfconset": {
"type": "multiple",
"description": "Concatenating a set $L$ with itself $n$ times is written $L^n$. So $L^1$ is just $L$. Concatenating a set zero times ($L^0$) will give the set $\\{\\lambda\\}$. To help undestand why this is true, remember that the $L^* = L^+ \\cup \\{\\lambda\\}$",
"question": "Let $\\Sigma = \\{a,b\\}$, $L = \\{ab, aba\\}$. What is $L^0$?",
"answer": ["$\\{\\lambda\\}$"],
"choices": ["$\\{\\lambda\\}$", "$\\{\\}$", "$\\lambda$"]
},
"setconx": {
"type": "multiple",
"description": "We can concatenate a set with itself some number of times. For set $L$ concatenated $n$ times, as you might expect, we write this $L^n$.",
"question": "For $L$={$ab$,$aba$}, what is $L^2$? ",
"answer": "$\\{abab, ababa, abaab, abaaba\\}$",
"choices": ["$\\{abab, ababa, abaab, abaaba\\}$", "$\\{ab, ababa, abaa, abaa\\}$", "$\\{ab, aba, abaab, abaaba\\}$", "$\\{a, b, abaab, abaaba\\}$", "$\\{a, ab, aba, abaaba\\}$"]
},
"setcon3": {
"type": "multiple",
"description": "We can concatenate a set with itself some number of times. For set $L$ concatenated $n$ times, as you might expect, we write this $L^n$.",
"question": "[T, F] For $L = \\{a, b\\}$, $L^3$ is all 3 letter combinations of $a$ and $b$.",
"answer": "True",
"choices": ["True", "False"]
}
}
}
}