cyto.json

    },
    "003-005": {
      "source": "003",
      "target": "005",
      "label": "A mapping induces images and preimages."
    },
    "005-006": {
      "source": "005",
      "target": "006",
      "label": "Injectivity and surjectivity can be expressed in terms of images and preimages."
    },
    "005-305": {
      "source": "005",
      "target": "305",
      "label": "Characterize continuity in terms of preimages of open balls."
    },
    "005-307": {
      "source": "005",
      "target": "307",
      "label": "Topological properties of images under continuous functions."
    },
    "110-307": {
      "source": "110",
      "target": "307",
      "label": "Continuous functions preserve compactness of sets in their image."
    },
    "111-307": {
      "source": "111",
      "target": "307",
      "label": "The Heine-Borel theorem gives the existence of maxima and minima for continuous functions defined on compact sets."
    },
    "307-506": {
      "source": "307",
      "target": "506",
      "label": "The existence of minima and maxima for continuous functions on compact sets is necessary for the proof of Rolle's theorem."
    },
    "000-006": {
      "source": "000",
      "target": "006",
      "label": "Image and Preimage are characterized via logical statements involving quantifiers."
    },
    "003-007": {
      "source": "003",
      "target": "007",
      "label": "Composition is an operation on maps."
    },
    "001-105": {
      "source": "001",
      "target": "105",
      "label": "Supremum and infimum are numbers associated to sets of real numbers."
    },
    "002-105": {
      "source": "002",
      "target": "105",
      "label": "Supremum and infimum are numbers associated to sets of real numbers."
    },
    "100-115": {
      "source": "100",
      "target": "115",
      "label": "Being Cauchy is a property of a sequence."
    },
    "115-106": {
      "source": "115",
      "target": "106",
      "label": "Completeness can be defined by stating that every Cauchy sequences converges."
    },
    "105-106": {
      "source": "105",
      "target": "106",
      "label": "Existence of suprema of bounded sets is a characterization of completeness."
    },
    "001-110": {
      "source": "001",
      "target": "110",
      "label": "Open, closed and compact are properties of sets of numbers."
    },
    "001-112": {
      "source": "001",
      "target": "112",
      "label": "The interior, the closure or the boundary of a set is a set again."
    },
    "002-100": {
      "source": "002",
      "target": "100",
      "label": "A sequence is a map having the real numbers as domain "
    },
    "002-101": {
      "source": "002",
      "target": "101",
      "label": "The definition of convergence involves the absolute value of a difference of real numbers and a quantitative comparison with another real number."
    },
    "002-106": {
      "source": "002",
      "target": "106",
      "label": "Real numbers are complete by the completeness axiom."
    },
    "003-100": {
      "source": "003",
      "target": "100",
      "label": "A sequence is a map of natural numbers "
    },
    "003-300": {
      "source": "003",
      "target": "300",
      "label": "Function is just another name for map."
    },
    "003-304": {
      "source": "003",
      "target": "304",
      "label": "Continuity is a central notion for maps on the real numbers."
    },
    "004-011": {
      "source": "004",
      "target": "011",
      "label": "A lot of identities for sums and products are proved via induction."
    },
    "004-100": {
      "source": "004",
      "target": "100",
      "label": "A sequence is a map having the natural numbers as domain."
    },
    "006-400": {
      "source": "006",
      "target": "400",
      "label": "The exponential function is a bijective function onto the positive reals."
    },
    "400-401": {
      "source": "400",
      "target": "401",
      "label": "The bijectivity of the exponential gives the logarithm function as inverse."
    },
    "206-400": {
      "source": "206",
      "target": "400",
      "label": "The exponential series is absolutely convergent."
    },
    "211-400": {
      "source": "211",
      "target": "400",
      "label": "The functional equation of the exponential is a consequence of the Cauchy product."
    },
    "006-505": {
      "source": "006",
      "target": "505",
      "label": "Bijective differentiable functions can be differentiated using the inversion formula."
    },
    "006-401": {
      "source": "006",
      "target": "401",
      "label": "The natural logarithm is the inverse of the exponential function."
    },
    "304-401": {
      "source": "304",
      "target": "401",
      "label": "The natural logarithem is the inverse of a continuous function."
    },
    "007-306": {
      "source": "007",
      "target": "306",
      "label": "Composition of continuous functions gives a continuous function."
    },
    "007-502": {
      "source": "007",
      "target": "502",
      "label": "How to differentiate a composition of functions."
    },
    "500-502": {
      "source": "500",
      "target": "502",
      "label": "Break down differentiability of compositions to the differentiability of simpler functions."
    },
    "100-101": {
      "source": "100",
      "target": "101",
      "label": "Having a limit is a property of a sequence."
    },
    "101-109": {
      "source": "101",
      "target": "109",
      "label": "Limit inferior and limit superior are in particular limits."
    },
    "107-109": {
      "source": "107",
      "target": "109",
      "label": "Limit inferior and limit superior are the largest and smallest accumulation points of a subsequence."
    },
    "101-103": {
      "source": "101",
      "target": "103",
      "label": "Calculating limits of sums, products or quotients of convergent sequences."
    },
    "101-104": {
      "source": "101",
      "target": "104",
      "label": "Monotonicity and boundedness imply convergence."
    },
    "101-117": {
      "source": "101",
      "target": "115",
      "label": "Every convergent sequence is also a Cauchy sequence."
    },
    "101-110": {
      "source": "101",
      "target": "110",
      "label": "Closedness is characterized by convergence of sequences."
    },
    "107-110": {
      "source": "107",
      "target": "110",
      "label": "Compactness is characterized by existence of converging subsequences."
    },
    "101-107": {
      "source": "101",
      "target": "107",
      "label": "A sequence may have a convergent subsequence and this limit is then an accumulation value of the original sequence."
    },
    "101-108": {
      "source": "101",
      "target": "108",
      "label": "Bolzano Weierstrass guarantees convergence of certain sequences."
    },
    "102-104": {
      "source": "102",
      "target": "104",
      "label": "A bound for the sequence is also a bound for the limit by the monotonicity of limits."
    },
    "102-108": {
      "source": "102",
      "target": "108",
      "label": "The Bolzano Weierstrass Theorem guarantees the existence of accumulation points for bounded sequences."
    },
    "103-203": {
      "source": "103",
      "target": "203",
      "label": "Considering series as limits of sequences of partial sums leads to limit theorems for sequences."
    },
    "000-108": {
      "source": "000",
      "target": "108",
      "label": "Bolzano Weierstrass theorem is a logical statement."
    },
    "108-109": {
      "source": "108",
      "target": "109",
      "label": "Bolzano Weierstrass theorem guarantees the existence of a limit inferior and superior."
    },
    "110-111": {
      "source": "110",
      "target": "111",
      "label": "Heine-Borel theorem characterizes bounded and closed sets."
    },
    "110-112": {
      "source": "110",
      "target": "112",
      "label": "Interior is an open set, closure and boundary are closed sets."
    },
    "111-112": {
      "source": "111",
      "target": "112",
      "label": "The closure of a bounded set is always compact by Heine-Borel theorem."
    },
    "104-111": {
      "source": "104",
      "target": "111",
      "label": "Sandwiching gives rise to the proof using nested intervals in the Heine-Borel theorem."
    },
    "104-309": {
      "source": "104",
      "target": "309",
      "label": "The convergence of the intervals is based on monotonicity of the underlying sequences."
    },
    "108-309": {
      "source": "108",
      "target": "309",
      "label": "The nested intervals technique used in the proof of the Bolzano-Weierstrass theorem is also used for the proof of the intermediate value theorem."
    },
    "107-108": {
      "source": "107",
      "target": "108",
      "label": "Bolzano Weierstrass guarantees existence of accumulation values."
    },
    "100-200": {
      "source": "100",
      "target": "200",
      "label": "A series is a sequence of partial sums."
    },
    "101-200": {
      "source": "101",
      "target": "200",
      "label": "A convergent series is a convergent sequence."
    },
    "011-200": {
      "source": "011",
      "target": "200",
      "label": "Partial sums are described using the formal sum notation."
    },
    "200-201": {
      "source": "200",
      "target": "201",
      "label": " "
    },
    "200-203": {
      "source": "200",
      "target": "203",
      "label": " "
    },
    "200-204": {
      "source": "200",
      "target": "204",
      "label": "The Cauchy criterion works for sequences of partial sums that are Cauchy."
    },
    "106-204": {
      "source": "106",
      "target": "204",
      "label": "Cauchy sequences converge, therefore the Cauchy criterion for series gives convergence of a series."
    },
    "115-204": {
      "source": "115",
      "target": "204",
      "label": "If the sequence of partial sums is a Cauchy sequence, then it converges."
    },
    "102-204": {
      "source": "102",
      "target": "204",
      "label": "The Cauchy criterion involves bounds on certain parts of the partial sum."
    },
    "008-204": {
      "source": "008",
      "target": "204",
      "label": "The necessary criterion for series convergence is an implication but no equivalence."
    },
    "203-205": {
      "source": "203",
      "target": "205",
      "label": "A sufficient criterion for convergence of a series."
    },
    "008-205": {
      "source": "008",
      "target": "205",
      "label": "The Leibniz criterion gives the sufficient condition for convergence of a series but no equivalence."
    },
    "200-205": {
      "source": "200",
      "target": "205",
      "label": "The Leibniz criterion makes a statement about partial sums of an alternating sequence."
    },
    "104-205": {
      "source": "104",
      "target": "205",
      "label": "The Leibniz criterion makes a statement about partial sums of an monotonic sequence."
    },
    "200-206": {
      "source": "200",
      "target": "206",
      "label": "A strong concept of convergence for sequences of partial sums."
    },
    "201-206": {
      "source": "201",
      "target": "206",
      "label": "The geometric series is an example of an absolutely convergent sequence."
    },
    "205-206": {
      "source": "205",
      "target": "206",
      "label": "The Leibniz theorem shows that the alternating harmonic series is convergent but not absolutely convergent."
    },
    "200-207": {
      "source": "200",
      "target": "207",
      "label": "The comparison tests compare the terms of the partial sums to the ones of other series."
    },
    "200-208": {
      "source": "200",
      "target": "208",
      "label": " "
    },
    "102-209": {
      "source": "102",
      "target": "209",
      "label": "If the nth root of the underlying sequence is bounded by a number striclty smaller than one, the series converges by the root criterion."
    },
    "200-210": {
      "source": "200",
      "target": "210",
      "label": "Reordering a series leads to a new sequence of partial sums."
    },
    "206-210": {
      "source": "206",
      "target": "210",
      "label": "Reordering an absolutely convergent sequence does not change the limit."
    },
    "203-210": {
      "source": "203",
      "target": "210",
      "label": "Reordering of a series may change the limit of the series."
    },
    "006-210": {
      "source": "006",
      "target": "210",
      "label": "Reordering of a series is defined via an bijective mapping of natural numbers."
    },
    "200-211": {
      "source": "200",
      "target": "211",
      "label": " "
    },
    "109-209": {
      "source": "109",
      "target": "209",
      "label": "The root criterion in its limit form gives a condition for the limsup of the underlying sequence."
    },
    "209-403": {
      "source": "209",
      "target": "403",
      "label": "The root criterion is useful for determining domains of convergence."
    },
    "201-209": {
      "source": "201",
      "target": "209",
      "label": "The geometric series is used in the proof of the root criterion as a majorant."
    },
    "201-207": {
      "source": "201",
      "target": "207",
      "label": "The geometric series is often used as a majorant in the comparison test."
    },
    "201-208": {
      "source": "201",
      "target": "208",
      "label": "The geometric series is used in the proof of the quotient criterion as a majorant."
    },
    "207-208": {
      "source": "207",
      "target": "208",
      "label": "The proof of the quotient criterion relies on majorant criterion."
    },
    "208-400": {
      "source": "208",
      "target": "400",
      "label": "By the quotient criterion on can show that the exponential series converges absolutely."
    },
    "207-209": {
      "source": "207",
      "target": "209",
      "label": "The proof of the root criterion relies on majorant criterion."
    },
    "206-208": {
      "source": "206",
      "target": "208",
      "label": " The quotient criterion is a statement about absolute convergence of a series."
    },
    "206-209": {
      "source": "206",
      "target": "209",
      "label": " The root criterion is a statement about absolute convergence of a series."
    },
    "206-211": {
      "source": "206",
      "target": "211",
      "label": "Taking the Cauchy product is an operation on two absolute convergent series."
    },
    "005-300": {
      "source": "005",
      "target": "300",
      "label": "Boundedness of a function can be expressed in terms of the image or range."
    },
    "100-300": {
      "source": "100",
      "target": "300",
      "label": "A sequence of functions gives rise to a sequence of real numbers via point evaluations."
    },
    "101-303": {
      "source": "101",
      "target": "303",
      "label": "Limits of functions are limits of sequences of real numbers."
    },
    "001-300": {
      "source": "001",
      "target": "300",
      "label": "Bounded functions on an interval form a set."
    },
    "300-301": {
      "source": "300",
      "target": "301",
      "label": "Pointwise convergence is a notion of convergence for sequences of functions."
    },
    "301-302": {
      "source": "301",
      "target": "302",
      "label": "Pointwise convergence is a weaker notion of convergence for sequences of functions than uniform convergence."
    },
    "300-308": {
      "source": "300",
      "target": "308",
      "label": "Properties of the limit of a sequence of continuous functions."
    },
    "301-308": {
      "source": "301",
      "target": "308",
      "label": "The pointwise limit of a sequences of continuous functions does not have to be continuous."
    },
    "300-400": {
      "source": "300",
      "target": "400",
      "label": "The exponential series can be considered as a sequence of partial sums of functions"
    },
    "200-400": {
      "source": "200",
      "target": "400",
      "label": "The exponential series can be considered as a sequence of partial sums of functions"
    },
    "300-403": {
      "source": "300",
      "target": "403",
      "label": "A power series is essentially a sequence of partial sums of functions."
    },
    "001-010": {
      "source": "001",
      "target": "010",
      "label": "We use different operations to work with multiple sets."
    },
    "010-110": {
      "source": "010",
      "target": "110",
      "label": "Set operations may be used to modify set properties like opennes, closedness or compactness."
    },
    "304-500": {
      "source": "304",
      "target": "500",
      "label": "Differentiability of a function is a stronger property than continuity."
    },
    "500-501": {
      "source": "500",
      "target": "501",
      "label": "How to differentiate a sum of differentiable functions."
    },
    "500-506": {
      "source": "500",
      "target": "506",
      "label": "Rolle's theorem makes a statement about zeros of the first derivative of a differentiable function."
    },
    "506-507": {
      "source": "506",
      "target": "507",
      "label": "A notion stronger than pointwise convergence is uniform convergence."
    },
    "300-302": {
      "source": "300",
      "target": "302",
      "label": "Special notion of convergence for sequences of functions."
    },
    "101-304": {
      "source": "101",
      "target": "304",
      "label": "Continuous functions map convergent sequences to convergent sequences."
    },
    "304-400": {
      "source": "304",
      "target": "400",
      "label": "The exponential function is continuous."
    },
    "103-306": {
      "source": "103",
      "target": "306",
      "label": "Continuity of sums, products, and quotients is preserved as a consequence of the limit theorems for sequences."
    },
    "304-306": {
      "source": "304",
      "target": "306",
      "label": "Continuity is preserved under certain operations with functions."
    },
    "502-609": {
      "source": "502",
      "target": "609",
      "label": "Integration by substitution is loosely speaking the inverse operation to calculating the derivative via the chain rule."
    },
    "304-307": {
      "source": "304",
      "target": "307",
      "label": "Continuity preverves compactness of sets."
    },
    "304-309": {
      "source": "304",
      "target": "309",
      "label": "Continuous functions on intervals don't have jumps. Instead they attain every value in between to elements of their range."
    },
    "303-304": {
      "source": "303",
      "target": "304",
      "label": "If a function is continuous at a point, the the value coincides with the function limit at this point."
    },
    "302-304": {
      "source": "302",
      "target": "304",
      "label": "Uniform convergence of continuous functions leads to a continuous limit function."
    },
    "305-308": {
      "source": "305",
      "target": "308",
      "label": "The epsilon-delta criterion allows for an elegant proof of the continuity of the uniform limit of a sequence of continuous functions."
    },
    "103-303": {
      "source": "103",
      "target": "303",
      "label": "Calculating limits of functions builds on the limit theorems of sequences."
    },
    "402-403": {
      "source": "402",
      "target": "403",
      "label": "Each partial sum of a power series is a polynomial."
    },
    "110-305": {
      "source": "110",
      "target": "305",
      "label": "Characterize continuity using open intervals."
    },
    "402-511": {
      "source": "402",
      "target": "511",
      "label": "Use polynomials to approximate differentiable functions."
    },
    "303-508": {
      "source": "303",
      "target": "508",
      "label": "Calculate limits of functions using l'Hospital's rule."
    },
    "403-511": {
      "source": "403",
      "target": "511",
      "label": "A Taylor series is a particular power series."
    },
    "508-509": {
      "source": "508",
      "target": "509",
      "label": "Further situations where limits can be calculated using l'Hospital's rule."
    },
    "403-503": {
      "source": "403",
      "target": "503",
      "label": "Uniform convergence within the domain of convergence of a power series."
    },
    "300-503": {
      "source": "300",
      "target": "503",
      "label": "Differentiability of sequences of functions."
    },
    "500-503": {
      "source": "500",
      "target": "503",
      "label": "Differentiability of sequences of functions."
    },
    "301-503": {
      "source": "301",
      "target": "503",
      "label": "Proving pointwise convergence of a sequence of differentiable functions is part of proving the differentiability of the limit function."
    },
    "302-503": {
      "source": "302",
      "target": "503",
      "label": "Proving uniform convergence of the derivatives of a function is part of proving the differentiability of the limit function."
    },
    "510-511": {
      "source": "510",
      "target": "511",
      "label": "Evaluations of higher derivatives form the coefficients of a Taylor polynomial."
    },
    "511-512": {
      "source": "511",
      "target": "512",
      "label": "Application of Taylor's theorem to a particular example."
    },
    "512-513": {
      "source": "512",
      "target": "513",
      "label": "Proof of Taylor's theorem."
    },
    "304-305": {
      "source": "304",
      "target": "305",
      "label": "A different notion of continuity using open balls."
    },
    "002-600": {
      "source": "002",
      "target": "600",
      "label": "Functions that are constant on intervals of real numbers."
    },
    "003-600": {
      "source": "003",
      "target": "600",
      "label": "Step functions are a particular type of piecewise defined mapping."
    },
    "600-601": {
      "source": "600",
      "target": "601",
      "label": "A first notion of area under a function graph for step functions."
    },
    "601-603": {
      "source": "601",
      "target": "603",
      "label": "Generalizing the concept of an integral to bounded functions via approximation."
    },
    "500-606": {
      "source": "500",
      "target": "606",
      "label": "Differentiation is in some sense the opposite of integration."
    },
    "500-606": {
      "source": "500",
      "target": "606",
      "label": "Differentiation is in some sense the opposite of integration."
    },
    "601-602": {
      "source": "601",
      "target": "602",
      "label": "The definition of the Riemann integral gives rise to nice properties."
    },
    "603-609": {
      "source": "603",
      "target": "609",
      "label": "A tool to calculate the integral of a bounded function."
    },
    "006-609": {
      "source": "006",
      "target": "609",
      "label": "Transformations of the domain of integration need to be invertible or at least injective."
    },
    "500-609": {
      "source": "500",
      "target": "609",
      "label": "Transformations of the domain of integration need to be differentiable."
    },
    "400-609": {
      "source": "400",
      "target": "609",
      "label": "Using the substitution rule may help to calculate integrals of exponentials."
    },
    "007-609": {
      "source": "007",
      "target": "609",
      "label": "Some integrands in the substitution rule involve compositions of functions."
    },
    "207-613": {
      "source": "207",
      "target": "613",
      "label": "Comparison principles aim at inheriting a nice property from a well-known object to a new object of study."
    },
    "606-607": {
      "source": "606",
      "target": "607",
      "label": "The second fundamental theorem tells us how antiderivatives differ."
    },
    "500-607": {
      "source": "500",
      "target": "607",
      "label": "The second fundamental theorem connects integration and differentiation."
    },
    "601-604": {
      "source": "601",
      "target": "604",
      "label": "By definition the Riemann integral consists of limits of integrals of step functions."
    },
    "603-605": {
      "source": "603",
      "target": "605",
      "label": "Properties of the Riemann integral."
    },
    "605-609": {
      "source": "605",
      "target": "609",
      "label": "The Riemann integral comes with an order of the integral boundaries."
    },
    "605-608": {
      "source": "605",
      "target": "608",
      "label": "A further property of the Riemann integral is the mean-value theorem."
    },
    "501-610": {
      "source": "501",
      "target": "610",
      "label": "This integration rule can be seen as an inversion of the product rule of differentiation."
    },
    "603-610": {
      "source": "603",
      "target": "610",
      "label": "A further important rule for calculating integrals."
    },
    "507-608": {
      "source": "507",
      "target": "608",
      "label": "A further mean value property for integrals instead of derivatives."
    },
    "603-604": {
      "source": "603",
      "target": "604",
      "label": "Compute integrals according to the definition."
    },
    "603-607": {
      "source": "603",
      "target": "607",
      "label": "The second fundamental theorem connects integration and differentiation."
    },
    "203-613": {
      "source": "203",
      "target": "613",
      "label": "Characterize the convergence of an integral by the convergence of a suitable series."
    },
    "612-613": {
      "source": "612",
      "target": "613",
      "label": "Estimate improper Riemann integrals via series."
    },
    "604-606": {
      "source": "604",
      "target": "606",
      "label": "The fundamental theorem of calculus allows to reuse examples of integrals and derivatives in order to calculate new integrals."
    },
    "603-612": {
      "source": "603",
      "target": "612",
      "label": "An improper integral is defined via a limit of Rimeann integrals on bounded sets."
    },
    "003-606": {
      "source": "003",
      "target": "606",
      "label": "An antiderivative is also a map."
    },
    "303-612": {
      "source": "303",
      "target": "612",
      "label": "An improper integral is defined as the limit of a function (the antiderivative)."
    },
    "003-402": {
      "source": "003",
      "target": "402",
      "label": "Polynomials can be interpreted as real valued functions."
    },
    "609-611": {
      "source": "609",
      "target": "611",
      "label": "The substitution rule is often used for calculating integrals of rational functions."
    },
    "108-106": {
      "source": "108",
      "target": "106",
      "label": "The fact that every Cauchy sequence converges can be derived from the Bolzano-Weierstraß theorem."
    },
    "102-117": {
      "source": "102",
      "target": "115",
      "label": "Cauchy sequences are bounded."
    },
    "402-611": {
      "source": "402",
      "target": "611",
      "label": "How to integrate rational functions."
    },
    "008-009": {
      "source": "008",
      "target": "009",
      "label": "How to construct logical statements."
    },
    "004-012": {
      "source": "004",
      "target": "012",
      "label": "Countable sets can be bijectively mapped to the natural numbers or a finite subset of the natural numbers."
    },
    "012-113": {
      "source": "012",
      "target": "113",
      "label": "Uncountability of the real numbers in particular means that the reals are not a countable set."
    },
    "104-113": {
      "source": "104",
      "target": "113",
      "label": "The uncountability of the reals can be shown via a interval nesting which converges do to the monotonicity of the underlying sequences."
    },
    "106-113": {
      "source": "106",
      "target": "113",
      "label": "Uncountability of the real numbers can be proven as a consequence of the completeness of the real numbers."
    },
    "006-012": {
      "source": "006",
      "target": "012",
      "label": "Countable sets can be bijectively mapped to the natural numbers or a finite subset of the natural numbers."
    },
    "203-207": {
      "source": "203",
      "target": "207",
      "label": "Characterize the convergence/divergence of a series by the convergence/divergence of a majorant/minorant."
    }
  }
}