definition 3.23 Predicate

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:613 · p. 23

ground object -- no derivation owed

Rests on

depends_on definition 3.1 Logical proposition ¶
depends_on definition 3.38 Ordered pair and Cartesian product ¶

Supports

depends_on definition 3.24 Quantifiers ¶
- depends_on definition 3.43 Map ¶
  - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.43 Group homomorphism ¶
    - depends_on definition 4.44 Group isomorphism ¶
      - depends_on definition 4.45 Group automorphism ¶
      - depends_on definition 5.143 Faithful representation ¶
      - depends_on proposition 4.70 Characterization of the direct product ¶
      - depends_on proposition 4.55 $S_3$ and the equilateral triangle ¶
      - depends_on theorem 4.56 Cayley's theorem ¶
    - depends_on definition 4.73 External semidirect product ¶
      - depends_on example 4.78 The Euclidean group ¶
      - depends_on example 4.79 The Poincaré group ¶
      - depends_on proposition 4.74 The semidirect product is a group ¶
      - depends_on proposition 18.20 The Galilean group ¶
      - depends_on theorem 4.76 Internal characterization of the semidirect product ¶
    - depends_on definition 5.141 Representation of a group ¶
      - depends_on definition 5.146 Character ¶
      - depends_on definition 5.142 Dimension of a representation ¶
      - depends_on definition 5.145 Equivalent representations ¶
      - depends_on definition 5.143 Faithful representation ¶ ↺
      - depends_on definition 5.147 Invariant subspace ¶
      - depends_on definition 5.144 Unitary representation ¶
    - depends_on proposition 4.42 When the homomorphisms form a group ¶
    - depends_on proposition 4.47 Image of the inverse element ¶
    - depends_on proposition 4.46 Image of the neutral element ¶
      - depends_on proposition 4.47 Image of the inverse element ¶ ↺
    - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
  - depends_on definition 4.16 Image ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.15 Absolute-value function ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.14 Inverse function ¶
      - depends_on proposition 7.31 Chain rule ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on definition 7.10 Constant function ¶
      - depends_on proposition 7.28 Derivative of a constant function ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 18.11 Acceleration ¶
      - depends_on definition 18.10 Velocity ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.19 Infinite limits ¶
      - depends_on definition 9.86 Outer and inner expansions; matching ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on remark 7.21 rem:ana-discontinuities ¶
      - depends_on remark 7.17 rem:ana-limit-punctured ¶
      - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
    - depends_on definition 7.12 Product of functions ¶
      - depends_on proposition 7.30 Leibniz rule ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on proposition 7.29 Linearity ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 7.140 Hausdorff measure ¶
      - depends_on definition 7.142 Box-counting dimension ¶
      - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
      - depends_on remark 7.144 What is developed here, and what is not ¶
    - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on definition 5.22 Metric associated with a norm ¶
    - depends_on definition 10.24 Well-posed problem ¶
      - depends_on corollary A.152 Uniqueness ¶
      - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
      - depends_on corollary 10.81 Uniqueness for the heat equation ¶
      - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 10.25 The backward heat problem is ill posed ¶
      - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on definition 12.32 Separable Hilbert space ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 6.29 Sequential compactness ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
    - depends_on definition 6.26 Open ball; metric topology ¶
      - depends_on definition A.76 Star-shaped set ¶
      - depends_on lemma 6.30 Lebesgue number ¶
      - depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
    - depends_on definition 6.29 Sequential compactness ¶ ↺
    - depends_on lemma A.505 A continuous partition of unity ¶
      - depends_on lemma A.506 Locality ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Logical proposition	declared	parts/02-mathematical-methods/01-logic-sets.tex:622
`depends_on`	→	Ordered pair and Cartesian product	declared	parts/02-mathematical-methods/01-logic-sets.tex:622
`depends_on`	←	Quantifiers	declared	parts/02-mathematical-methods/01-logic-sets.tex:638