r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notebook/notes/set/cardinality.md

4.5 KiB
Raw Blame History

title TARGET DECK FILE TAGS tags
Cardinality Obsidian::STEM set::cardinality
set

Overview

We say set A is equinumerous to set B, written (A \approx B) if and only if there exists a functions#Injections function from A functions#Surjections B.

%%ANKI Basic Suppose A is equinumerous to B. How does Enderton denote this? Back: A \approx B Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic What does it mean for A to be equinumerous to B? Back: There exists a bijection between A and B. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Suppose A \approx B. Then what must exist? Back: A bijection between A and B. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Suppose there exists a one-to-one function F from A into B. When is A \approx B? Back: When F is also onto B. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Suppose there exists a function F from A onto B. When is A \approx B? Back: When F is also one-to-one. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Suppose there exists a one-to-one function F from A onto B. When is A \approx B? Back: Always, by definition. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

Equivalence Concept

For any sets A, B, and C:

  • A \approx A;
  • if A \approx B, then B \approx A;
  • if A \approx B and B \approx C, then A \approx C.

Notice though that \{ \langle A, B \rangle \mid A \approx B \} is not an equivalence relation since the equivalence concept of equinumerosity concerns all sets.

%%ANKI Basic Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory. Back: For all sets A, B, and C:

  • A \approx A;
  • A \approx B \Rightarrow B \approx A;
  • A \approx B \land B \approx C \Rightarrow A \approx C Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory. Back: Class \{ \langle A, B \rangle \mid A \approx B \} is reflexive on the class of all sets, symmetric, and transitive. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic What is the reflexive property of equinumerosity in FOL? Back: \forall A, A \approx A Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic What is the symmetric property of equinumerosity in FOL? Back: \forall A, B, A \approx B \Rightarrow B \approx A Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic What is the transitive property of equinumerosity in FOL? Back: \forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Is \{ \langle A, B \rangle \mid A \approx B \} a set? Back: No. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Why isn't \{ \langle A, B \rangle \mid A \approx B \} a set? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Is \{ \langle A, B \rangle \mid A \approx B \} an equivalence relation? Back: No. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Basic Why isn't \{ \langle A, B \rangle \mid A \approx B \} an equivalence relation? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

Bibliography

  • Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).