definition 3.62 Equipotence

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:1628 · p. 36

ground object -- no derivation owed

Rests on

depends_on definition 3.47 Bijective map ¶
depends_on definition 3.40 Cardinal ¶

Supports

depends_on definition 3.65 Countable set ¶
- depends_on corollary 3.68 The rationals are countable ¶
  - depends_on corollary A.3 The continuum is the power set of the naturals ¶
  - depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶
  - depends_on remark 7.94 Three numbers, two kinds ¶
- depends_on definition 7.138 Set of measure zero ¶
  - depends_on definition 7.140 Hausdorff measure ¶
    - depends_on definition 7.142 Box-counting dimension ¶
      - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
      - depends_on remark 7.144 What is developed here, and what is not ¶
    - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
    - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
  - depends_on lemma 7.139 A countable union of null sets is null ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
    - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
    - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
  - 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 proposition 3.67 Countability of the plane of naturals ¶
  - depends_on corollary 3.68 The rationals are countable ¶ ↺
  - depends_on lemma 7.139 A countable union of null sets is null ¶ ↺
  - depends_on lemma A.6 Lindenbaum ¶
  - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
- depends_on theorem 3.69 Uncountability of the continuum ¶
  - depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶ ↺
  - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
depends_on proposition 3.63 prop:inf-equipotence-equivalence ¶
depends_on theorem 3.72 Cantor–Schröder–Bernstein ¶

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`	→	Bijective map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1632
`depends_on`	→	Cardinal	declared	parts/02-mathematical-methods/01-logic-sets.tex:1632
`depends_on`	←	Countable set	declared	parts/02-mathematical-methods/01-logic-sets.tex:1665
`depends_on`	←	prop:inf-equipotence-equivalence	declared	parts/02-mathematical-methods/01-logic-sets.tex:1637
`depends_on`	←	Cantor–Schröder–Bernstein	declared	parts/02-mathematical-methods/01-logic-sets.tex:1849