R1846 Cardinality of set complement
R2220 Binary set intersection is commutative
R4148 Intersection is a lower bound to each set in the intersection
R4174 Proper contraction has at most a single fixed point
R4126 Almost surely bounded random complex number has all absolute moments finite
R1193 Finite product of indicator functions equals indicator of intersection
R4602 Element in finite cartesian product iff components in images of canonical projections
R3461 Holomorphic function with vanishing derivative on complex domain is constant
R3503 Data processing inequality
R4919 Measurable transformation preserves independent countable random collection
R1558 Second fundamental theorem of complex integral calculus
R1077 Maximum element is unique
R4654 Probability calculus expression for basic real conditional expectation given a complement partition of the sample space
R2556 Probability calculus expression for probability conditioned on event of nonzero probability
R4290 Vector space always has a linearly independent subset
R347 Isotonicity of map image
R4508 Two basic real numbers equal iff difference equals zero
R2522 Basic real conditional expectation given minimal information
R4794 Independent minimums preserve exponential distribution
R4904 Strong derivative for basic real square function
R4848 Probability of event conditional on complement
R112 Cavalieri principle
R2833 Nonempty set is singleton iff all elements equal each other
R3869 Row-standardisation of row-independent random basic real standard triangular array
R3060 Basic Riemann integral of even function over an interval symmetric abour zero
R2914 Cantor-Bernstein theorem
R2960 Function odd part is odd
R4269 Countable set union with empty set
R4518 Equivalent characterisations of finiteness of basic real variance
R2081 Binary set intersection with empty set is empty
R322 Image of projection map
R4038 Binary product of finite basic real sums
R4370 Set of euclidean natural numbers has Lebesgue measure zero
R3983 Subcollection of independent collection of sigma-algebras is independent
R3494 Glivenko-Cantelli theorem
R5093 Total number of fixed-length sequences using a given number of labels
R80 Lipschitz map is continuous
R3923 Standard gaussian random basic real number is symmetric about zero
R1064 Isotonicity of Lebesgue measure
R2719 Affine majorant for the standard natural logarithm
R4846 Conditional simple entropy with respect to itself
R4991 Complex matrix multiplied from the right by a diagonal matrix
R1832 Cardinality of a finite cartesian product of finite sets
R4899
R4805 Dual probability distribution function for geometric random basic positive integer
R4168 Difference of set and countable intersection equals union of differences
R5074 Determinant of a lower triangular complex matrix
R5049
R3757 Linearity of basic real matrix trace
R4438 Equivalent characterisations of ergodicity for probability-preserving system
R4049 Number of binary relations on a finite set
R4889 Basic random variable almost surely finite if expectation finite
R3589 Random basic number almost surely finite if expectation is finite
R3200 Characteristic function of Bernoulli random boolean number
R4765 Reflection preserves euclidean real subset relation
R3919 Probability measure need not be an injection
R4503 Countable union of sets of measure zero is of measure zero
R4903 Strong derivative for affine basic real function
R2016 Probabilistic Markov's inequality
R4746 Superadditivity and subadditivity of limit superior and limit inferior for unsigned basic integral
R4591 Binary additivity of transpose for basic real matrices
R371 Countable set iff injection to natural numbers
R233 Whole space is closed
R3958 Intersecting subtrahend with minuend preserves set difference
R4398
R2767 Identity map is an injection
R4145 Binary union is an upper bound to both sets in the union
R4961
R4303 Nonempty finite poset contains a maximal element
R3695 Standardised Lindeberg central limit theorem
R4470 Inverse image of countable union is union of inverse images
R468 Lagrange's theorem
R4755 Inverse image of reflected basic real set
R4699
R3192 I.I.D. sequence of standard gaussian random basic real numbers is standard gaussian random euclidean real number
R4902 Strong derivative for constant basic real function
R2413 Complex conjugate of real number
R4667 Basic real function which is strongly differentiable only at a single point
R4702
R708 Reverse triangle inequality for metric
R4692
R3897 Distribution of random finite euclidean real sum is convolution of distribution measures
R4780 Basic real conditional expectation given independent sigma-algebra
R4125 Two random real numbers identical in distribution have identical absolute moments
R4788
R976 Finite disjoint additivity of unsigned basic measure
R5066 Inverse of symmetric complex matrix is symmetric
R4444 Superset differenced from set equals empty set
R1960 Characterisation of measure-preserving transformations by signed integral
R3515 Unsigned basic integral zero iff function almost everywhere zero
R4992 Complex matrix multiplied from the left by a diagonal matrix
R4773 Inclusion-exclusion lower bound to probability of ternary intersection
R1089 Characterisation of convergent sequences in metric space
R4606
R3204 Characteristic function of geometric random positive integer
R310 Right-invertible map iff surjection
R4213 Countable set union is invariant under bijective shifting of indices
R2754 Lipschitz map is Hölder continuous
R2093 Countable subadditivity of probability measure
R4164 Superset of union iff superset of every set in the union
R2782 Sum of initial natural numbers
R3304 Approximating sequence for the natural exponential function
R4464
R3984 Subcollection of independent random collection is independent
R3860 Quantile function is inverse to invertible distribution function for random basic real number
R2079 Isotonicity of set intersection
R2981 Riemann integral is particular case of Lebesgue integral
R2966 Indicator function under scaling of the argument
R88 Convolution is associative
R2641 Euclidean real martingale is constant in expectation
R4293 Power set of a finite set is finite
R4950 Real matrix with no real eigenvalues
R4275 Absolute value function is an even basic real function
R3532 Tautological pointwise lower bound from indicator function on sublevel set
R4093 Basic real harmonic and arithmetic means are multiplicative inverses when arguments are
R3911 Probability density function for standard gaussian random basic real number
R49 Isotonicity of subtracting same set
R3183 First fundamental theorem of Riemann integral calculus
R4467 Empty set is a stationary measurable set
R1160 Reflection invariance of Euclidean volume function
R1540 Affine map preserves convex sets in images
R3507 Binary union of countable sets is countable
R2343 Uncorrelated basic real weak law of large numbers
R714 Element belongs to its own equivalence class
R4776 Hoeffding's lemma
R4278 Expression for probability of event in terms of complement event
R2370 Basic real arithmetic expression for cardinality of countable set
R5072 Determinant of triangular complex matrix
R3652 Cramér-Wold theorem
R4802 Law of total probability for complement partition in terms of random variables
R5111 Number of injections is proportional to number of subsets of given size
R4801 Probability mass function for cogeometric random basic natural number
R4274 Absolute value function escapes to positive infinity in both directions
R4996
R1034 Power set is closed under unions
R2929 Sum of odd functions is odd
R1530 Inverse image preserves set difference
R2092 Sequential continuity of probability measure from above
R4330 Probability of finite intersection with an almost sure event
R4011 Polish space is second-countable
R797 Principle of weak mathematical induction on the natural numbers
R3942 Derivative of gaussian basic real density function
R354 Image of singleton set
R4349 Two disjoint events independent iff one is of probability zero
R2764 Canonical singleton map is bijection
R4748
R2373 Second probabilistic Borel-Cantelli lemma
R2788 Real binomial theorem
R50 Set difference equals intersection with complement
R4895 Standard counting measure is a point-mass measure
R25 Cauchy-Schwarz inequality
R237 Intersection distributes over union
R2780 Upper bound for number of arrows in finite digraph
R2783 Sum of initial natural numbers squared
R3531 Pointwise product with indicator function is lower bound for unsigned basic function
R4580 Empty set union equals the empty set
R3543 Endpoint bounds for basic real affine combination
R4264 Cartesian product with empty set is empty
R4163 Subset of binary intersection iff subset of both sets in the intersection
R4834
R4206 Basic real golden ratio is a root of a quadratic basic real polynomial
R2310 Distribution formula for unsigned basic integral
R2112 Isotonicity of Riemann integral
R3412 Euclidean real arithmetic mean minimises Euclidean distance
R4932
R5076 Bijection from the natural number plane to the set of natural numbers
R4817 Data processing inequality for transformed endpoint
R3646 Countable partition additivity of probability measure
R5068 Determinant of scaled complex matrix
R2368 I.I.D. strong law of large numbers for random real numbers
R4207
R4463 Strongly mixing probability-preserving system is weakly mixing
R3922 Gaussian random basic real number is symmetric about its expectation
R4871 Translation property of the standard log-sum-exp function
R4774
R3694 Lindeberg central limit theorem
R4855 Reflection property of standard logarithm function
R4804 Probability distribution function for geometric random basic positive integer
R3617 Turning a finite number of unsigned basic real numbers into convex combination coefficients
R4308 Map composition need not be commutative
R1242 Unsigned basic integral is compatible with measure
R3904 Characteristic function of a Poisson random natural number
R2067 Subtracting empty set from set
R4583
R4639 Characteristic function of translated random basic real number
R2537 Basic real exponentiation function with positive exponent is strictly isotone on positive basic reals
R1851 Cardinality of set of permutations on finite set
R3493 Empirical distribution function converges in probability to the true distribution function
R4905 Strong derivative for standard natural basic real logarithm function
R4483 Basic real number is zero iff equal to its reflection
R4295 Power set of a countable set need not be countable
R4982 Exchangeable random collection is identically distributed
R1074 Rolle's theorem
R4831 Homomorphism property of standard logarithm function
R4329 Probability of countable intersection with an almost sure event
R4769
R4559 Probability of complement of an almost sure event
R4733 Conditional expectation of known random euclidean real number
R2182 Isotonicity of basic real sublevel sets
R4431
R4698 Unsigned basic integral with respect to a point measure
R4906
R2441 Reflection invariance of Lebesgue outer measure
R4732
R1070 Extreme value theorem for basic real calculus
R3383 Stein's lemma
R4615 Sufficient conditions to qualify as an approximate identity to complex Lebesgue convolution
R4152
R2687 Approximating function for the natural exponential function
R4541 Open set is its own interior
R220 Difference of set and union equals intersection of differences
R3863 Signed basic integral with respect to a point measure
R3529 Tautological pointwise inequality from indicator functions on sublevel sets
R4331 Probability of binary intersection with an almost sure event
R4502 Difference of almost everywhere equal euclidean complex functions is almost everywhere zero
R1177 Constant map is always measurable
R4825 Bit entropy of a standard bernoulli number is one bit
R3852 Transforming random basic real number to uniform random basic real number
R1575 Goursat's theorem for rectangles
R4777 Expectation of bounded random basic real number is within the bounding interval
R4287 Finite product of even complex functions is even
R1818 Isotonicity of basic real expectation
R2876 Basic mean value inequality
R4538 Inclusion of image does not imply inclusion of inverse image
R2091 Sequential continuity of probability measure from below
R4866 Generating a random basic real number with randomness from a standard uniform variable
R463 Law of the excluded middle is a propositional tautology
R2741 Set equality iff inclusion in both directions
R2599 Cantor's theorem
R4853 Tight lower bound to simple mutual information
R2464 Determinant is preserved under transpose of complex matrix
R5009 Bernoulli random boolean integer from a uniform random basic real number
R447 Zorn's lemma
R5081 Complex matrix determinant zero iff some nonzero vector is mapped to zero
R2436 Polar representation of complex conjugate
R4767
R3205 Probability mass function for geometric random positive integer
R755 Group centre is Abelian group
R4267 Set intersection with empty set is empty
R4088 Peirce basic boolean logic gate iff complement of maximum function on basic boolean ordered pairs
R3778 Probability density function for exponential random positive basic real number
R4272 Absolute value function is not strictly antitone
R4676
R4140 Standardised Mill's inequalities
R590 Square root of two is irrational
R975 Isotonicity of unsigned basic measure
R2658 Asymptotic results for standard logarithm function
R4368 Set of euclidean integers has Lebesgue measure zero
R3642 Law of total probability for complement partition
R5069 Determinant of diagonal complex matrix
R4165 Superset of countable union iff superset of every set in the union
R4142 Union is an upper bound to each set in the union
R2262 Basic real variance partition into first and second moments
R3972 Expression for random basic real sum of squares in terms of sample mean and variance
R3517 Unsigned basic expectation zero iff random number almost surely zero
R3595 Probabilistic Tonelli's theorem
R4703
R1904 Isotonicity of finite basic real summation
R2367 Law of total variance
R2100 Triangle inequality for signed basic expectation
R4573 Complex expectation over a null event is zero
R5097 Power set is isomorphic to a set of boolean functions
R3974 Sum of independedent standard chi-squard random basic real numbers is chi-squared
R4598 Standard gaussian basic real density function is an even function
R4630 Bijection between parenthesis-sliced cartesian triple products
R5086 Eigenvectors of a symmetric real matrix are orthogonal
R4286 Standard natural basic real logarithm function maps one to zero
R3582 Signed expectation finite iff absolutely integrable random basic number
R4671 First-degree polynomial approximation for a standard basic real exponential function near zero
R4823 Conditional probability of the empty event
R298 Identity map is continuous if topology on domain set is equal or stronger
R3640 Conditional probability given independent sigma-algebra
R4151 Binary intersection is a lower bound to each set in the intersection
R3544 Endpoint bounds for basic real arithmetic mean
R4970
R4687 Additivity of variance for a finite number of independent random real numbers
R4486 Metric space is Hausdorff
R508 Finite cartesian product of countable sets is countable
R796 Principle of weak mathematical induction
R3605 Subadditivity of basic real square root function
R1204 Fatou's lemma for unsigned basic integral
R4009 Metrisable topological space is Fréchet
R4714 Sufficient condition for the existence of density function for a random euclidean real number
R301 Canonical identity map is an injection
R4348 Two disjoint events are not necessarily independent
R4087 Real arithmetic expressions for OR boolean logic gate
R4261 Basic real binomial theorem for exponent two
R3262 Dilations of unsigned tail probability converge to zero
R4674
R2062 Antitonicity of subtracting from the same set
R3912 Characteristic function of rademacher random basic integer
R5099 Sum of initial natural numbers cubed
R2411 Characteristic function of exponential random basic real number
R2656 De Moivre-Laplace theorem
R4912 Strong derivative for symmetric euclidean real quadratic form
R4390 Fourier transform characterises euclidean real probability measure
R3583 Random basic number absolutisation equals sum of positive and negative parts
R2440 Translation invariance of Lebesgue outer measure
R3038 Closed graph theorem
R1487 Ring is a subring of itself
R246 Cauchy sequence is bounded
R4601 Element in countable cartesian product iff components in images of canonical projections
R4808 Affine transformations preserve independent basic real pairs
R2410 I.I.D. basic real weak law of large numbers under finite second absolute moments
R4374 Integral factorisation on a binary product of basic real lebesgue measure spaces
R4851 Tight upper bound to simple entropy
R4372 Independent random basic real pair is uncorrelated
R4943 Basic real binomial theorem for exponent three
R270 Basic real arithmetic expression for sum of finite basic real geometric sequence
R4898
R4520 Complex conditional expectation is absolutely integrable
R4847 Probability of event conditional on itself
R5075 Complex matrix determinant is multiplicative
R3908 Characteristic function of almost surely constant random euclidean real number
R3896 Expectation is idempotent
R4720 Conditional probability of complement event
R293 Open set less closed set is open
R4455
R3747 Transpose of finite product of finite basic real matrices
R2506 Independent random real collection is uncorrelated
R4701
R4339 Conditional probability of almost surely false event
R4187 Complement of an F-sigma set is a G-delta set
R1975 Measure of set in backward orbit under measure-preserving endomorphism
R4688 Second moment upper bound to basic real variance
R4604 Density partition for natural number moment of random basic real number
R4711 Probability of complement of event which is not the whole sample space need not be nonzero
R4189
R1903 Basic integral of almost everywhere zero function is zero
R2407 Measurable transformation preserves independence
R2928 Finite sum of even euclidean real functions is even
R3970 Measure of finite union finite iff measure of all sets in union finite
R4686 Probabilistic Chebyshov's inequality for an exponentiation function
R1076 Minimum element is unique
R4350 Finite superadditivity of finite set cardinality
R4656 Probability calculus expression for conditional probability given a countable partition of the sample space
R4456
R2747 Finite sets are closed in Hausdorff space
R4844 Mutual information of a simple random variable with respect to itself
R3179 Basic real binomial inequality
R2270 Sigma-algebra is closed under symmetric differences
R4739 Binary subadditivity of probability measure
R4327 Probability of countable union with an almost sure event
R4369 Set of euclidean rational numbers has Lebesgue measure zero
R2989 Unsigned basic Lebesgue integral under scaling
R4814 Markov chain triple iff reverse is a Markov chain triple
R4273 Absolute value function preserves zero
R4383 Euclidean real Fourier transform is uniformly bounded
R248 Binary subadditivity of basic real square root function
R2959 Function even part is even
R1370 Multilinear map is zero if any argument is zero
R2483 Variance of Bernoulli random boolean number
R4563 Complex integral over a set of measure zero is zero
R4596 Basic real calculus expression for moments of standard gaussian random basic real number
R2588 Weak Stirling formula
R977 Ambient set is union of subset and complement of subset
R4265 Finite set intersection with empty set is empty
R4649 Probability calculus expression for basic real conditional expectation given a countable partition of the sample space
R3198 Finite sum of uncorrelated Poisson random natural numbers is Poisson
R3513 Measurable subnull set has measure zero
R4843 Conditional entropy formula for simple mutual information
R3498 Homomorphism property of the standard natural logarithm
R4915
R800 Proof by principle of weak mathematical induction
R2732 Kolmogorov zero-one law
R1213 Linearity of unsigned basic integral
R4838 Sum rule for simple marginal probability
R4473 Almost idempotency of conditional expectation of random complex number
R2690 Minimum and maximum of stopping times are stopping times
R3749 Decomposition of basic real square matrix into sum of symmetric and antisymmetric parts
R1234 Borel-Cantelli lemma
R2615 Empty set is cofinite within a finite set
R1839 Cardinality of power set of a finite set
R1164 Translation invariance of Lebesgue measure
R4280 Standard natural basic real exponential function is strictly isotone
R4048 Number of binary relations on binary cartesian product
R4183 Alternative basic real arithmetic expressions for Catalan sequence
R4770 Inclusion-exclusion lower bound to probability of binary intersection
R4169 Difference of set and finite intersection equals union of differences
R3601 Gaussian approximation to standard Poisson distribution
R1603 De Moivre's theorem
R4729 Complex binomial inequality
R4627
R3748 Finite basic real matrix is positive definite iff symmetric part is
R457 Fréchet space iff singletons are closed
R4012 Polish space is Hausdorff
R244 Convergent sequence is Cauchy
R893 Cyclic group is Abelian
R2751 Fourier transform of absolutely integrable function is bounded
R4741 Probabilistic Chebyshov's inequality for square function
R648 Proportionally bounded linear map is Lipschitz
R4171 Difference of set and countable union equals intersection of differences
R4127 Basic real exponentiation function with unsigned exponent is isotone on unsigned basic reals
R3745 Basic real square matrix antisymmetric part is zero definite
R4211 Basic real golden ratio is limit of successive terms in the Fibonacci natural number sequence
R4737
R3514 Unsigned basic expectation over set of probability zero is zero
R4304 Nonempty finite poset contains a minimal element
R2972 The four classes of basic real intervals are each closed under dilation
R4276 Triangle inequality for absolute value function
R4925 Probability calculus expression for conditional probability given disjoint non-null partition
R3472 Maximum modulus principle
R4283 Standard natural basic real logarithm function escapes to positive infinity at positive infinity
R4159 Power set is not empty
R2589 Trivial factorial upper bound
R1824 Birkhoff ergodic theorem
R4321 Set and set complement are disjoint
R3231 Riemann integral analogue to infinite geometric series
R10 Binary cartesian product with empty set is empty
R4704
R4500
R4916
R4260 Basic real square root function is strictly isotone
R4693
R1959 Young's inequality
R1178 Continuous map is Borel measurable
R3649 Expectation of conditional probability
R2746 Finite sets are closed in Fréchet space
R4626 Strict isotonicity of basic Riemann integral
R5008 Random unsigned basic number almost surely finite if expectation is finite
R1012 Monotonicity of real limits
R2219 Set intersection is associative
R47 Set difference is subset of the minuend
R1401 Orthogonality relation is a symmetric binary relation
R5073 Determinant of an upper triangular complex matrix
R1232 Tonelli's theorem for sums and integrals
R4288
R3680 Characteristic function of standard gaussian random real number
R4659 Empirical probability distribution function is an unbiased estimator of the true distribution function
R292 Union is smallest upper bound
R4909 Strong derivative for self-dot product function
R3770 Product of finite basic real 2-by-2 matrix with its transpose
R4166 Superset of finite union iff superset of every set in the union
R4089 Real arithmetic expressions for NAND boolean logic gate
R4338 Conditional probability of almost surely true event
R1236 Markov's inequality
R4568 Countable indicator partition of a random complex number
R2141 Reflection invariance of unsigned basic Lebesgue integral
R4118 Basic real arithmetic expression for unsigned basic real geometric mean
R2259 Variance of a finite sum of random basic real numbers
R5080 Complex matrix determinant nonzero iff kernel is trivial
R4540 Inverse map is a bijection
R4787
R4340 Bayes' theorem in the case of event and complement
R3385 Standard approximating sequence for the natural exponential function
R4447
R4800 Number of injections from a finite set to itself
R4132 Strict version of probabilistic Markov inequality
R4143 Countable union is an upper bound to each set in the union
R2897 Fourier transform under scaling
R4013 Polish space is Fréchet
R4215 Countable set intersection is invariant under bijective shifting of indices
R3898 Distribution of sum of two random euclidean real numbers is convolution of distribution measures
R75 Intersection of closed sets is closed
R3518 Partition of random basic number into positive and negative parts
R108 Inclusion iff reverse inclusion of complements
R4609
R2748 Finite sets are closed in metric space
R1406 Collection of sets ordered by inclusion contains a maximal element
R4618 Riemann integral of standard basic real gaussian function on the real line
R2082 Binary set union with empty set
R2437 Euclidean real Fourier inversion theorem
R1869 Characterisation of equality of maps
R4631 Cardinality of finite cartesian products is invariant under insertion of parentheses
R3542 Jensen's inequality for finite basic real arithmetic mean
R1170 Euclidean volume of singleton
R4668 Transpose of finite product of finite complex matrices
R4147 Not all subsets of a cartesian product need be cylinder sets
R1856 Cardinality of the set of maps between finite sets
R81 Bilipschitz map is continuous
R507 Empty set is countable
R4689 Probabilistic Cavalieri principle
R5085 Eigenvalues of a symmetric real matrix are real-valued
R2938 Vanishing sequence is necessary for finiteness of real series
R4973 Bias-variance partition of mean square error for random basic real number
R2525 Independence of sigma-algebras is hereditary
R3429 Complex conjugation is multiplicative operation
R4162 Subset of finite intersection iff subset of every set in the intersection
R4469 Countable union of stationary measurable sets is stationary
R2986 Lebesgue outer measure under scaling
R4001 Whole space is clopen
R3508 Fermat's last theorem
R979 Countable subadditivity of measure
R4706 Complex conjugate zero iff argument zero
R4282 Standard natural basic real exponential function vanishes at negative infinity
R4684 I.I.D. weak law of large numbers for random real numbers
R4901 Probability density function for standard exponential random positive basic real number
R2086 Finite inclusion-exclusion principle for unsigned basic measure
R4738 Finite subadditivity of probability measure
R3020 Complex Lebesgue integral under scaling
R822 Group of prime order is cyclic
R1073 Mean value theorem
R2313 Expectation with dual probability distribution function
R4150 Finite intersection is a lower bound to each set in the intersection
R1854 Cardinality of the set of injections between finite sets
R4341 Bayes' theorem in the case of two pullback events
R1214 Isotonicity of unsigned basic integral
R4610
R3260 Weak law of large numbers for random real triangular arrays
R3506 Finite union of countable sets is countable
R4277 Measure of measurable set complement
R1237 Unsigned basic Borel function almost everywhere finite if integral is finite
R4757 Young's inequality for two basic real numbers
R3746 Transpose of binary product of finite basic real matrices
R4695 Infinite product of basic real numbers on the open unit interval need not converge to zero
R4333 Binary product of indicator functions equals indicator of intersection
R4170 Difference of set and binary intersection equals union of differences
R1171 Isotonicity of Euclidean volume function
R2786 Complement property of binomial coefficient
R1130 Intersection is largest lower bound
R1131 Image of inverse image
R4443 Measure of symmetric difference of set and subset
R4921 Affine transformations preserve independent countable basic real collection
R3783 Law of total probability for complex expectation in terms of pullback events
R4279 Explicit algebraic expression for elements of generated subgroup with singleton generator set
R4445 Inclusion-exclusion principle for probability of binary union
R5104 Proof by principle of weak mathematical induction on the natural numbers
R5092 Probability of binary intersection with a null event
R4947 Variance of standard Bernoulli random boolean number
R4537 Inclusion of inverse image does not imply inclusion of image
R24 Parallelogram identity
R2744 Square root of prime is irrational
R1841 Indicator function operator is a bijection
R4795 Basic real calculus expression for exponential distribution function
R4073 Probability of union with impossible event
R4944 Basic real binomial theorem for exponent four
R4811 Markov chain triple iff endpoints conditionally independent
R1450 Explicit algebraic expression for elements of generated subgroup
R3935 Superset iff equal to union with subset
R4182 Isotonicity of sublevel probabilities for random basic real number
R4466 Whole space is a stationary measurable set
R4086 Real arithmetic expressions for AND boolean logic gate
R1202 Dominated convergence theorem for signed basic integral
R3504 Erdős discrepancy problem
R4948 Gaussian approximation to standard binomial distribution
R2871 Equivalent characterisations of a Schwartz function
R2970 The four classes of basic real intervals are each closed under translation
R4809 Affine transformations preserve independent finite basic real collection
R3545 Endpoint bounds for basic real convex combination
R3404 Bayes' theorem in the case of two events
R1165 Reflection invariance of Lebesgue measure
R2089 Unsigned basic expectation is compatible with probability measure
R4511 Two complex numbers distinct iff difference distinct from zero
R2090 Isotonicity of probability measure
R4149 Countable intersection is a lower bound to each set in the intersection
R4597 Centred gaussian basic real density function is an even function
R2964 Even function equals its even part
R1488 Group is a subgroup of itself
R3479 Conformal complex function iff holomorphic bijection
R4292 Finite cartesian product of finite sets is finite
R3641 Conditional probability given independent random variable
R3207 Probability density function for a chi-squared random unsigned basic real number
R2224 Set union is associative
R2076 Inverse image of empty set
R2075 Image of empty set
R1369 Jensen's inequality for expectation
R2963 Odd function equals its odd part
R4836 Change of base formula for simple entropy
R1906 Isotonicity of limits of basic real sequences
R4607 Pushforward change of variables formula for unsigned basic integral
R4536 Map image empty iff underlying set empty
R4679 Independent countable random basic real collection is uncorrelated
R4914 Derivative for affine real matrix function
R3954 Two almost surely equal random variables are identically distributed
R4172 Difference of set and finite union equals intersection of differences
R431 Set of integers is countable
R5091 Lindeberg central limit theorem for a standard triangular array
R4214 Finite set union is invariant under bijective shifting of indices
R3621 Standard natural exponential function equals powers of Napier's constant
R3568 Basic real arithmetic-geometric mean inequality
R4965 Relationship with the sum of initial natural numbers and the binomial coefficient
R3416 Absolute value bounds for a basic real number
R4328 Probability of finite union with an almost sure event
R3939 Upper and lower bounds for codomain set of finite unsigned basic measure
R1158 Translation invariance of Euclidean volume function
R4876
R4727 Euclidean real binomial inequality
R4637
R978 Measure of set difference
R2740 Splitting a Riemann integral over an implicit interval partition
R370 Countable set iff surjection from natural numbers
R982 Sequential continuity of measure from below
R3215 Characteristic function of indicator random boolean number
R3920 Sublevel events do not necessarily coincide for identically distributed random variables
R4971
R1557 Basic real multiplicative-additive convex combination inequality
R4294 Set difference with finite minuend is finite
R125 Subset relation is reflexive
R1902 Signed basic integral of almost everywhere equal functions
R4781 Conditional expectation of known random basic real number
R1022 Partition of basic function into positive and negative parts
R4972 Bias-variance partition of mean square error for random basic real column matrix
R262 Countable union of countable sets is countable
R2160 Conditional expectation of known random complex number
R4728
R4560 Probability of complement of a null event
R4616 The standard mollifier satisfies sufficient conditions required of an approximate identity
R3612 Standard natural basic real logarithm function is strictly isotone
R4926 Finite partition additivity of unsigned basic measure
R5102 Count of boolean functions on initial positive integers with given sum
R4000 Empty set is clopen
R4740
R4090 Real arithmetic expressions for XOR boolean logic gate
R4158 Power set is multiplicative
R4566 Countable indicator partition of a complex function
R2405 Characteristic function identifies distribution of random basic real number
R4888 Isotonicity of set cardinality
R4764
R3516 Signed basic expectation of almost surely zero random number is zero
R4657 Probability calculus expression for conditional probability given a finite partition of the sample space
R4599 Density partition for expectation of random basic real number
R4325 Probability one if conditional probability one relative to event of probability one
R4157 Superadditivity of power set
R2623 Indicator function operator on set of N-subsets is bijection
R1159 Euclidean volume function under scaling
R4314 Number of boolean functions on a finite set
R1075 Basic real interior extremum theorem
R4291 Vector space always has an inclusion-maximal linearly independent set
R4430 Probability of event in backward orbit under probability-preserving endomorphism
R1838 Cardinality of finite set union of finite sets
R4857 Standard logarithm of a positive basic real number raised to an integer power
R4743 Riemann integral average of vanishing unsigned basic real function vanishes
R2485 Expectation of geometric random basic positive integer
R4920 Measurable transformation preserves independent finite random collection
R4997
R2164 Law of the excluded middle in probability calculus
R4907 Strong derivative for constant euclidean real function
R465 De Morgan's laws are propositional tautologies
R4387 Characteristic function of standard Poisson random natural number
R4666 Basic real geometric-harmonic mean inequality
R4509 Two basic real numbers distinct iff difference distinct from zero
R87 Convolution is commutative
R4543 Map inverse is invertible
R2806 Standardising a gaussian random real number
R4141 Probabilistic Chernoff inequality
R4485 Convergent sequence in metric space has unique limit point
R4665 Basic real weighted GM-HM inequality
R4010 Polish space is first-countable
R4167 Superset of binary union iff superset of both sets in the union
R1577 Cauchy's theorem for a disc
R4782 Expectation of basic real conditional expectation
R1035 Power set is closed under intersections
R315 Composition of surjections is surjection
R4945 Basic real binomial theorem for exponent five
R4660 Real empirical probability distribution function converges pointwise to the true distribution function
R3971 Intersection of sets of infinite measure need not have infinite measure
R4060 Appropriately parenthesised cartesian products are isomorphic
R4611
R4128 Two random real numbers identical in distribution have identical moments
R4826 Logarithm of a ratio
R2377 Almost sure convergence implies convergence in probability
R4453 Probability of symmetric difference of event and subevent
R4747
R102 Singletons are closed in a metric space
R4621 Standard mollifier is an approximate identity for complex Lebesgue convolution
R464 Ex falso sequitur quodlibet
R3367 Characterisation of convergence of geometric basic real series
R1973 Map composition is associative
R2221 Set intersection is invariant under bijective shifting of indices
R3401 Probability calculus expression for conditional expectation given disjoint non-null partition
R4697 Simple integral of a measurable simple function with respect to a point measure
R2934 Isotonicity of unsigned basic real series
R1275 Every map to trivial topological space is continuous
R5060 Product of real matrix and adjugate is a constant diagonal matrix
R4869 Characterisation of superconvex basic real functions
R432 Set of natural numbers is countable
R4495 Superconvex and superconvex basic real functions on open basic real interval are continuous
R4173 Difference of set and binary union equals intersection of differences
R1368 Jensen's inequality
R4317 Inclusion-exclusion principle for unsigned basic measure of ternary union
R4092 Basic real arithmetic expression for basic real harmonic mean
R4332
R2484 Moments of Bernoulli random boolean number
R1120 Antitonicity of minimum
R7 Empty set is subset of every set
R4662 Signed basic expectation with respect to a point probability measure
R984 Transforming an isotone sequence of sets into pairwise disjoint sequence with common limit
R4315 Cardinality of a finite set raised to a finite power
R4595 Basic real calculus expression for moments of centred gaussian random basic real number
R1166 Lebesgue measure under scaling
R3901 Domain of map to monoid equals union of kernel and support
R1868 Composition of indicator function with set endomorphism
R3201 Characteristic function of a binomial random natural number
R2505 Basic real covariance partition into first moments
R511 Natural number plane is countable
R4886 Characterisation of a quotient set by proper set partitions
R2975 Lebesgue sigma-algebra is closed under reflections, translations, and scaling
R3 Stirling formula
R4669 Diagonalizable complex matrix is symmetric
R981 Countably infinite measurable cover has measurable subcover
R4289 Vector space always has a basis
R4827 Entropy of a standard bernoulli number
R3581 Absolute moment inherits finiteness from greater exponents for random complex number
R4510 Two complex numbers equal iff difference equals zero
R2222 Set union is invariant under bijective shifting of indices
R831 Second group isomorphism theorem
R4930
R5059 Adjugate for 2-by-2 real matrix
R255 Power set of empty set
R4685 I.I.D. weak law of large numbers for random real numbers of finite first absolute moments
R4822 Conditional probability of the sample space
R2203 Expectation of poisson random basic natural number
R4577 Countable indicator partition of complex expectation in terms of pullback events
R3931 Subtracting set from empty set
R3471 Open complex mapping theorem
R1338 Probabilistic Borel-Cantelli lemma
R4730 Riemann integral of a constant function on the closed unit interval
R3789 Absolute value strictly less than real number iff in symmetric open interval
R3946 Interior is an open set
R4512
R2150 Expectation of conditional expectation
R4603
R4908 Strong derivative for affine euclidean real function
R4208
R1848 Bijection between parenthesis-sliced cartesian products
R5045 Basic real matrix multiplied by column matrix from the right
R4326 Probability of binary union with almost sure event
R4271 Absolute value function is not strictly isotone
R4513
R3944 Open set iff equal to interior
R3618 Turning two unsigned basic real numbers into convex combination coefficients
R2950 Complex conjugation operation is an involution
R4873
R3679 Standardised I.I.D. real central limit theorem
R4661 Integer power recurrence of basic real golden ratio
R2138 Translation invariance of unsigned basic Lebesgue integral
R2787 Pascal's rule
R1233 Finite additivity of unsigned basic integral
R4678 Independent finite random basic real collection is uncorrelated
R4783
R5063 Cofactor partition of determinant for real square matrix
R4139 Mill's inequalities
R4378 Independent random basic real pair iff distribution functions factorise
R517 Singletons are closed in Hausdorff space
R2060 Probability of set difference
R1544 Anova partition of orthogonal projection
R399 Injectivity is hereditary
R1744 Finite union of finite sets is finite
R4074 Arbitrary intersection of subsets is subset
R3559 Probability of random basic real number taking value in right-closed interval
R2548 Constant map pulls back a bottom sigma-algebra
R1566 Mean value theorem for Riemann integral
R2535 Chebyshov's inequality
R4562 Basic integral over a set of measure zero is zero
R4391 Characteristic function identifies distribution of random euclidean real number
R4927 Binary partition additivity of unsigned basic measure
R4929 Binary partition additivity of probability measure
R4281 Standard natural basic real exponential function is escapes to positive infinity at positive infinity
R4829 Base inversion property of standard natural logarithm function
R4144 Finite union is an upper bound to each set in the union
R219 Difference of set and intersection equals union of differences
R4841 Finite disjoint additivity of probability measure
R2120 Bounded continuous function is Riemann integrable on a basic real interval
R983 Sequential continuity of measure from above
R3232 Value of standard logarithm function at its parameter value
R4095 Basic real arithmetic expression for basic real arithmetic mean
R3535 Riemann integral of unit semicircle
R3676 Characteristic function for sum of independent random euclidean real numbers
R4323
R4468 Set of stationary measurable sets is a sigma-algebra
R4091 Biconditional boolean logic gate iff complement of trivial distance function on basic boolean ordered pairs
R5094 Number of boolean functions on a boolean cartesian product
R4262 Finite cartesian product with empty set is empty
R3259 Weak law of large numbers for variance with weighted decay
R4534
R2875 Cauchy's basic real mean value theorem
R4437 Complement of stationary measurable set is stationary
R2113 Riemann integral of a constant function
R4830 Change of base formula for logarithm function
R3843 I.I.D. real central limit theorem
R4754
R1162 Lebesgue sigma-algebra is closed under reflections
R76 Empty set is closed
R3988 Expression for quadratic form of finite basic real square matrix in terms of symmetric part
R2466 Real matrix determinant is multiplicative
R2539 Probabilistic Chebyshov's inequality
R5061 Expression of real matrix inverse in terms of adjugate
R4161 Subset of countable intersection iff subset of every set in the intersection
R4931
R4648 Union of sigma-algebras need not be a sigma-algebra
R2223 Binary set union is commutative
R560 Division theorem in the ring of basic integers
R756 Abelian group iff centre is whole group
R1276 Every map from discrete topological space is continuous
R3776 Probability to win an exponential race
R1831 Real arithmetic expression for binomial coefficient
R3061 Basic Riemann integral over a reflected interval
R1149 Every point in open set is an interior point
R3774 Probability distribution function of exponential random basic number
R4673 Napier's constant equals a limit of products with factors nearing one
R4320 Set of singletons is subset of power set
R1514 Isotonicity of signed basic integral
R3587 Random basic number almost surely finite iff positive and negative parts are
R3938 Upper and lower bounds for codomain set of probability measure
R2933 Isotonicity of convergent basic real series
R4310 Number of N-ary relations on a finite set
R4632 Cardinality of cartesian triple products is invariant under insertion of parentheses
R716 Equivalence class is not empty
R3969 Measure of intersection finite if measure of at least one set finite
R2757 Proper contraction map is continuous
R4263 Countable cartesian product with empty set is empty
R4663 Basic real square root function is subconvex
R1372 Inner product is zero if either argument is zero
R738 Left and right cosets coincide in Abelian group
R3586 Basic function is finite almost everywhere iff positive and negative parts are
R4268 Finite set union with empty set
R3500 Homomorphism property of the standard natural basic real exponential function
R263 Binary cartesian product of countable sets is countable
R4796 Probability for two independent exponential random basic real numbers to be smaller than the other
R3325 Banach fixed point theorem
R4638 Complex-linearity of complex matrix trace
R4228 Basic real ordering is compatible with addition
R4216 Finite set intersection is invariant under bijective shifting of indices
R1743 Intersection of finite sets is finite
R4756
R4672 First-degree polynomial approximation for a standard natural basic real exponential function near zero
R3228 Bounded sequence need not be Cauchy
R4887 Cardinality of binary cartesian product of finite sets
R4567 Countable indicator partition of a random euclidean real number
R3995 Complex binomial theorem
R4998 Limit of distribution function of geometric random basic positive integer scaled by reciprocal of index
R3166 Fourier transform of absolutely integrable function vanishes at infinity
R1371 Bilinear map is zero if either argument is zero
R3562 Basic real conditional variance partition into conditional moments
R1194 Indicator function with respect to set complement
R4007 Singletons are closed in Polish space
R2595 Convolution identity for binomial coefficient
R4565 Countable indicator partition of a euclidean real function
R828 First group isomorphism theorem
R4832 Homomorphism property of standard logarithm function in the binary case
R2665 Characteristic function of gaussian random real number
R2014 Triangle inequality for signed basic integral
R4675
R477 Set finiteness is hereditary
R355 Inverse image of singleton set
R466 Principle of double-negation is a propositional tautology
R176 Canonical set epimorphism is surjection
R4112 Strong fundamental theorem of complex algebra
R2653 Basic real conditional expectation is absolutely integrable
R4617 The standard mollifier integrates to one
R26 Pythagorean theorem
R4928 Finite partition additivity of probability measure
R5079 Characterisation of complex matrix eigenvalues in terms of characteristic polynomial
R221 Cartesian product is not associative
R4845 Joint entropy formula for simple mutual information
R4055 Singleton map is injection from set to power set
R4849 Symmetry of simple mutual information
R2422 Complex-linearity of Fourier transform
R4188 Set cardinality in terms of finite ambient set
R3184 Second fundamental theorem of Riemann integral calculus
R2074 Isotonicity of inverse image
R90 Complex function convolution is homogeneous to degree one
R4763 Set complement of map inverse image
R2435 Every complex number aside from zero has polar form
R2395 Subset of empty set is empty
R2745 The only subset of finite set with same cardinality is the ambient set itself
R3645 Countable partition additivity of unsigned basic measure
R1036 Power set is closed under complements
R309 Left-invertible map iff injection
R3885 Gaussian approximation to Poisson distribution
R1030 Sigma-algebra is closed under countable intersections
R4628 Map to empty set is a surjection
R4309 Number of N-ary relations on a finite cartesian product
R2295 Unit increment property of the gamma function
R3956 Expectation of random unsigned basic number by integrating tail probabilities
R4334 Every nonempty finite graph contains a bipartite subgraph with at least half the edges
R3953 Two almost surely equal random basic real numbers are identically distributed
R4285 Standard natural basic real exponential function maps zero to one
R4270 Set union with empty set
R3600 Finite sum of independent Poisson random natural numbers is Poisson
R3440 Uniformly continuous map preserves Cauchy sequences
R3719 Probability of complement event
R3945 Set is a superset to its interior
R4117 Endpoint bounds for basic real linear combination
R352 Inverse image of image
R4913 Strong derivative for euclidean real quadratic form without translation
R4222 Quotienting a set reduces its cardinality
R1565 Goursat's theorem
R1176 Inverse image of a constant map is either empty or equal to the domain set
R4786 Basic real conditional covariance partition into conditional moments
R3384 Components of gaussian random euclidean real number are gaussian random basic real numbers
R3289 Sequence in product space is Cauchy iff each component sequence is Cauchy
R4489 Sigma-algebra is closed under limit superiors and limit inferiors
R3844 Characteristic function of scaled random basic real number
R4900
R30 Inverse image of intersection is intersection of inverse images
R1564 Oriented complex curve integral of continuous function with a primitive on open set
R3462 Goursat's theorem for circles
R5077 Injection from the natural number plane to the set of natural numbers
R2158 Conditional expectation given independent sigma-algebra
R3921 Identically distributed random collection need not be stationary
R4079 Modus tollendo ponens in propositional logic
R4461
R1650 Interior point of set iff has open neighbourhood contained in set
R3960 Identically distributed random variables need not be almost surely equal
R3206 Probability mass function for binomial random natural number
R2254 Riemann integral of pristine basic real gaussian function on the real line
R3955 Components of standard gaussian random euclidean real number are standard gaussian random basic real numbers
R4989 Information inequality
R4160 Subset of intersection iff subset of every set in the intersection
R4186 Complement of a G-delta set is an F-sigma set
R3784 Law of total probability for complex expectation in terms of pullback events of a discrete random variable
R5046 Absolute moment inherits finiteness from greater exponents for random real number
R4705 Inner product with repeating argument is a basic real number
R3883 Row-standardisation of row-independent random basic real triangular array
R2071 Isotonicity of set union
R4670 Basic real geometric mean is a right limit of basic real power means
R4266 Countable set intersection with empty set is empty
R3316 Pointwise limit of continuous functions need not be continuous
R2337 Density function for binary sum of independent random basic real numbers
R368 Image of countable set is countable
R2677 Law of total probability for a countable partition of events of positive probability
R4318 Inclusion-exclusion principle for unsigned basic measure of binary union
R2168 Isotonicity of Lebesgue outer measure
R3415 Necessary and sufficient condition for convergence of geometric complex series
R3512 Unsigned basic integral over set of measure zero is zero
R2055 Cavalieri principle for basic real Riemann integral calculus
R424 Euler's identity
R2386 Characterisation of almost sure convergence using limit superior and limit inferior
R4680 I.I.D. strong law of large numbers for random real numbers of finite fourth moments
R1762 Closed set less open set is closed
R4867 Initial sum of powers of two
R4219 Dirichlet's theorem
R2374 Borel-Cantelli zero-one law
R4131 Markov lower bound on unsigned basic expectation
R4284 Standard natural basic real logarithm function escapes to negative infinity at zero
R17 Bijective map from natural numbers to integers
R4542 Map is inverse to its inverse
R2080 Union is upper bound for intersection
R1885 Absolute value of basic function equals sum of positive and negative parts
R3602 Gaussian approximation to binomial distribution
R1083 Minimal element is minimum element in ordered set
R4322 Cardinality of finite set union of disjoint sets
R2339 Finite sum of uncorrelated basic real gaussians is basic real gaussian
R74 Finite union of closed sets is closed
R2750 Fourier transform of finite unsigned euclidean real Borel measure is uniformly bounded
R4791 Probability for two independent standard Bernoulli random boolean numbers to coincide