Set Theory
A set is a collection of objects, considered as a whole.
Referenced by (24 direct, 124 transitive)
Direct references:
- Element
- Membership criterion
- Domain of Definition
- Value
- Range
- Sequence
- Inverse Image
- Vector Space
- countable-closed-set-has-isolated-points
- proof-of-theorem-27
- proof-of-subsequential-limits-of-a-metric-space-form-a-closed-set
- subsequential-limits-of-a-metric-space-form-a-closed-set
- note-31
- Diameter
- diameter-of-set-equals-diameter-of-closure
- proof-of-euclidean-spaces-are-complete
- Upper bound
- Lower bound
- Domain
- Cauchy-Riemann Equations
- Component
- Differentiable
- Total derivatives are unique
- Continuously Differentiable
Transitive (depth 1):
- proof-of-compact-metric-spaces-are-complete
- Vector
- Scalar
- Diverge
- Convergent
- Term
- Cauchy Sequence
- sequence-range-cardinality
- sequence-terms-not-distinct
- Subsequence
- sequence-notation
- every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- limit-of-a-function-characterized-by-limits-of-sequences
- Bolzano-Weierstrass
- Bounded (sequence)
- sequence-theorems-context
- limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- nested-sequence-of-compact-sets-with-lim-diam-zero-has-singleton-intersection
- sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- Limit (Sequence)
- limit-point-implies-convergent-sequence
- proof-of-limit-of-a-function-characterized-by-limits-of-sequences
- note-49
- real-sequence-notation
- mapping-continuous-iff-inverse-images-of-open-sets-are-open
- remark-7
- Vector Field
- Scalar Function
- Vector Function
- note-11
- Function
- Greatest lower bound
- proof-of-theorem-19
- remark-32
- Least upper bound
- note-4
- Cauchy's Integral Theorem
- Volume Integral
- Taylor Expansion Theorem
- Divergence Theorem of Gauss
- theorem-19
- Total Derivative
- gradient-as-surface-normal-vector
- Level Surface
- remark-9
- Laplacian
- Range (sequence)
- grad-div-curl-related
- proof-of-@cauchys-integral-theorem
- remark-46
- Partial Derivative
- Jacobian Matrix
- Directional Derivative
Transitive (depth 2):
- directional-derivative-is-inner-product-of-vector-and-grad
- Normal Derivative
- remark-45
- Surface Integral over Vector Field
- Stoke's Theorem
- Line Integral of Vector Function
- proof-of-limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- compact-metric-spaces-are-complete
- proof-of-theorem-50
- Cauchy criterion for convergence
- euclidean-spaces-are-complete
- Complete
- proof-of-theorem-23
- proof-of-sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-cauchy-criterion-for-convergence
- note-3
- Vector Multiplication by a Scalar
- Hessian Matrix
- Tangent Plane
- Direction
- remark-16
- Surface Normal Vector
- invariance-of-curl
- Bound vector
- Unit Vector
- note-5
- Free vector
- Layer
- intuition-13
- incompressible
- Cross Product
- proof-of-limit-of-a-function-at-a-point-is-unique-if-it-exists
- proof-of-theorem-7
- convergent-sequences-are-bounded
- Irrotational
- Potential Function
- Divergence
- Gradient
- Subsequential limit
- remark-30
- proof-of-function-is-continuous-at-point-iff-limit-at-point-equals-function-at-point
- remark-3
- Complex Function
- Composition
- Homeomorphism
- Contour Integral
- Circle of Convergence
- gravitational-potential-is-a-solution-to-laplaces-equation
Transitive (depth 3):
- Zero Vector
- Vector Equality
- Analytic
- Point Singularity
- Derivative Function
- Analytic at a point
- Complex Derivative
- analytic-implies-cr-equations
- proof-of-gradient-as-surface-normal-vector
- remark-23
- theorem-50
- example-52
- proof-of-integral-of-one-over-z-around-unit-circle
- Surface Normal
- Tangent Vector
- Radius of Convergence
Transitive (depth 4):
The objects that make up a set are called its elements or its members.
Referenced by (4 direct, 123 transitive)
Direct references:
Transitive (depth 1):
- remark-45
- remark-30
- proof-of-function-is-continuous-at-point-iff-limit-at-point-equals-function-at-point
- remark-3
- Sequence
- Complex Function
- note-11
- Hessian Matrix
- remark-7
- Domain of Definition
- Composition
- Jacobian Matrix
- remark-9
- Value
- Homeomorphism
- Range
- Continuously Differentiable
- Contour Integral
- Circle of Convergence
- Layer
- Component
- Direction
- remark-16
- Surface Normal Vector
- invariance-of-curl
- Bound vector
- Unit Vector
- note-5
- Free vector
- intuition-13
- incompressible
- Cross Product
- theorem-19
- note-3
- Vector Multiplication by a Scalar
- remark-32
Transitive (depth 2):
- Zero Vector
- directional-derivative-is-inner-product-of-vector-and-grad
- remark-46
- proof-of-theorem-19
- Normal Derivative
- Vector Equality
- Directional Derivative
- Diverge
- Convergent
- Term
- note-31
- Cauchy Sequence
- sequence-range-cardinality
- sequence-terms-not-distinct
- Subsequence
- sequence-notation
- every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- limit-of-a-function-characterized-by-limits-of-sequences
- Bolzano-Weierstrass
- Bounded (sequence)
- sequence-theorems-context
- proof-of-theorem-27
- limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- nested-sequence-of-compact-sets-with-lim-diam-zero-has-singleton-intersection
- sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- Limit (Sequence)
- subsequential-limits-of-a-metric-space-form-a-closed-set
- limit-point-implies-convergent-sequence
- proof-of-limit-of-a-function-characterized-by-limits-of-sequences
- note-49
- real-sequence-notation
- Analytic
- Point Singularity
- Derivative Function
- Analytic at a point
- Complex Derivative
- analytic-implies-cr-equations
- Vector Field
- Scalar Function
- Vector Function
- proof-of-gradient-as-surface-normal-vector
- gradient-as-surface-normal-vector
- remark-23
- Range (sequence)
- grad-div-curl-related
- proof-of-@cauchys-integral-theorem
- proof-of-integral-of-one-over-z-around-unit-circle
- Radius of Convergence
- Tangent Vector
- Partial Derivative
Transitive (depth 3):
- Divergence Theorem of Gauss
- Surface Integral over Vector Field
- Stoke's Theorem
- Line Integral of Vector Function
- proof-of-limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- proof-of-euclidean-spaces-are-complete
- compact-metric-spaces-are-complete
- proof-of-theorem-50
- Cauchy criterion for convergence
- euclidean-spaces-are-complete
- proof-of-compact-metric-spaces-are-complete
- Complete
- proof-of-theorem-23
- proof-of-sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-subsequential-limits-of-a-metric-space-form-a-closed-set
- proof-of-cauchy-criterion-for-convergence
- proof-of-analytic-implies-cr-equations
- Complex Differentiable
- theorem-7
- A theorem about the uniqueness of Taylor series as @power-series expansions
- A theorem about radius of convergence
- proof-of-limit-of-a-function-at-a-point-is-unique-if-it-exists
- proof-of-theorem-7
- Cauchy's Integral Theorem
- Cauchy-Riemann Equations
- Taylor Expansion Theorem
- Residue Theorem
- convergent-sequences-are-bounded
- Irrotational
- Potential Function
- Divergence
- Gradient
- Subsequential limit
Transitive (depth 4):
Two sets are equal if and only if they have the same elements. Formally, for any sets $A$ and $B$:
$$A = B \iff (\forall x)(x \in A \iff x \in B)$$
Referenced by (1 direct)
Direct references:
If $X$ and $Y$ are sets such that every element of $X$ is also an element of $Y,$ then we say $X$ is a subset of $Y,$ denoted as $X \subset Y.$ Formally,
$$X \subset Y \iff (\forall x)(x \in X \implies x \in Y)$$
Referenced by (7 direct, 6 transitive)
Direct references:
Transitive (depth 1):
Transitive (depth 2):
If $X$ and $Y$ are sets such that every element of $Y$ is also an element of $X,$ then we say $X$ is a superset of $Y,$ denoted as $X \supset Y.$ This is the same as $Y \subset X.$
Theorem
@set-equality-via-subset-inclusion
Two sets $X$ and $Y$ are equal if and only if $X$ is a subset of $Y$ and $Y$ is a subset of $X.$
Proof
@proof-of-set-equality-via-subset-inclusion
Suppose $X$ and $Y$ are sets with $X \subset Y$ and $Y \subset X.$ Now, suppose $x \in X.$ Then, $x \in Y.$ Conversely, suppose $y \in Y.$ Then $y \in X.$ Thus, $(\forall x)(x \in X \iff x \in Y),$ and $X = Y$.
$\square$Consider two sets, $A$ and $B,$ whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, any element of $B,$ which we denote by $f(x).$ Then $f$ is said to be a function from $A$ to $B.$
Referenced by (25 direct, 86 transitive)
Direct references:
- Random Variables and Probability Distributions
- Machine Learning Basics
- Layer
- Domain of Definition
- Value
- Range
- Sequence
- note-11
- proof-of-function-is-continuous-at-point-iff-limit-at-point-equals-function-at-point
- Composition
- Classification of Singularities
- Derivatives of Complex Functions
- remark-3
- Contour Integral
- Complex Function
- Circle of Convergence
- Component
- remark-7
- remark-9
- Continuously Differentiable
- remark-30
- Hessian Matrix
- Jacobian Matrix
- remark-45
- Homeomorphism
Transitive (depth 1):
- Diverge
- Convergent
- Term
- note-31
- Cauchy Sequence
- sequence-range-cardinality
- sequence-terms-not-distinct
- Subsequence
- sequence-notation
- every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- limit-of-a-function-characterized-by-limits-of-sequences
- Bolzano-Weierstrass
- Bounded (sequence)
- sequence-theorems-context
- proof-of-theorem-27
- limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- nested-sequence-of-compact-sets-with-lim-diam-zero-has-singleton-intersection
- sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- Limit (Sequence)
- subsequential-limits-of-a-metric-space-form-a-closed-set
- limit-point-implies-convergent-sequence
- proof-of-limit-of-a-function-characterized-by-limits-of-sequences
- note-49
- real-sequence-notation
- Analytic
- Point Singularity
- Derivative Function
- Analytic at a point
- Complex Derivative
- analytic-implies-cr-equations
- Vector Field
- Scalar Function
- Vector Function
- proof-of-theorem-19
- remark-32
- Range (sequence)
- grad-div-curl-related
- proof-of-@cauchys-integral-theorem
- proof-of-integral-of-one-over-z-around-unit-circle
- Radius of Convergence
- remark-46
- Partial Derivative
- Directional Derivative
Transitive (depth 2):
- directional-derivative-is-inner-product-of-vector-and-grad
- Normal Derivative
- Divergence Theorem of Gauss
- Surface Integral over Vector Field
- Stoke's Theorem
- Line Integral of Vector Function
- proof-of-limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- proof-of-euclidean-spaces-are-complete
- compact-metric-spaces-are-complete
- proof-of-theorem-50
- Cauchy criterion for convergence
- euclidean-spaces-are-complete
- proof-of-compact-metric-spaces-are-complete
- Complete
- proof-of-theorem-23
- proof-of-sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-subsequential-limits-of-a-metric-space-form-a-closed-set
- proof-of-cauchy-criterion-for-convergence
- proof-of-analytic-implies-cr-equations
- Complex Differentiable
- theorem-7
- A theorem about the uniqueness of Taylor series as @power-series expansions
- A theorem about radius of convergence
- proof-of-limit-of-a-function-at-a-point-is-unique-if-it-exists
- proof-of-theorem-7
- Cauchy's Integral Theorem
- Cauchy-Riemann Equations
- intuition-13
- Taylor Expansion Theorem
- Residue Theorem
- convergent-sequences-are-bounded
- Irrotational
- Potential Function
- Divergence
- Gradient
- gradient-as-surface-normal-vector
- theorem-19
- Subsequential limit
Transitive (depth 3):
Referenced by (6 direct, 17 transitive)
Direct references:
If $f$ is a function from the set $A$ to the set $B,$ the elements $f(x) \in B$ are called the values of $f.$
Referenced by (3 direct)
Referenced by (1 direct, 4 transitive)
Direct references:
Transitive (depth 1):
Transitive (depth 2):
Referenced by (27 direct, 18 transitive)
Direct references:
- Machine Learning Basics
- sequence-notation
- Term
- sequence-terms-not-distinct
- Sequences in Euclidean and Metric Spaces (embedded)
- Convergent
- Limit (Sequence)
- Diverge
- sequence-range-cardinality
- Bounded (sequence)
- sequence-theorems-context
- limit-point-implies-convergent-sequence
- Subsequence
- sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-theorem-27
- Bolzano-Weierstrass
- subsequential-limits-of-a-metric-space-form-a-closed-set
- note-31
- Cauchy Sequence
- limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- nested-sequence-of-compact-sets-with-lim-diam-zero-has-singleton-intersection
- proof-of-every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- every-convergent-sequence-in-a-metric-space-is-a-cauchy-sequence
- note-49
- proof-of-limit-of-a-function-characterized-by-limits-of-sequences
- limit-of-a-function-characterized-by-limits-of-sequences
- real-sequence-notation
Transitive (depth 1):
- proof-of-limit-of-diameter-of-remaining-points-in-cauchy-sequence-is-zero
- proof-of-euclidean-spaces-are-complete
- compact-metric-spaces-are-complete
- proof-of-theorem-50
- Cauchy criterion for convergence
- euclidean-spaces-are-complete
- proof-of-compact-metric-spaces-are-complete
- Complete
- proof-of-theorem-23
- proof-of-sequence-in-compact-metric-space-has-a-convergent-subsequence
- proof-of-subsequential-limits-of-a-metric-space-form-a-closed-set
- proof-of-cauchy-criterion-for-convergence
- proof-of-limit-of-a-function-at-a-point-is-unique-if-it-exists
- proof-of-theorem-7
- convergent-sequences-are-bounded
- Subsequential limit
Transitive (depth 2):
Note
@sequence-notation
If $f(n) = x_n,$ for $n \in J,$ it is customary to denote the sequence $f$ by the symbol $\{x_n\},$ or sometimes by $x_1, x_2, x_3, \dots.$
If $f: A \to B$ and $E \subset(B),$ then $f^{-1}(E)$ denotes the set of all $x \in A$ such that $f(x) \in E.$ We call $f^{-1}(E)$ the inverse image of $E$ under $f.$
Referenced by (1 direct)
Direct references:
De Morgan's Laws
Theorem
@complement-of-union-is-intersection-of-complements
The complement of a union is equal to the intersection of complements.
Proof
@proof-of-complement-of-union-is-intersection-of-complements
Let $A$ and $B$ be sets. We want to show that
$$ (A \cup B)^c = A^c \cap B^c. $$
Suppose $x \in (A \cup B)^c.$ Then, if $x \in A$ or $x \in B,$ then $x \in A \cup B$ and $x \notin (A \cup B)^c,$ a contradiction. Therefore, $x \notin A$ and $x \notin B.$ That is, $x \in A^c$ and $x \in B^c,$ therefore $x \in A^c \cap B^c.$
$\square$Referenced by (2 direct)
Theorem
@complement-of-intersection-is-union-of-complements
The complement of an intersection is equal to the union of complements.
Proof
@proof-of-complement-of-intersection-is-union-of-complements
Let $A$ and $B$ be sets. We want to show that
$$ (A \cap B)^c = A^c \cup B^c. $$
Suppose $x \in (A \cap B)^c.$ Then, $x$ is not in $A \cap B,$ that is, $x$ is either not in $A$ or it is not in $B$ or it is in neither. If $x$ is not in $A,$ then it is in $A^c,$ and therefore it is in $A^c \cup B^c.$ The same approach works with $B,$ and therefore $x \in A^c \cup B^c,$ and we have shown $(A \cap B)^c = A^c \cup B^c.$
$\square$