Download Notes on Cardinality and Equivalence Relations and more Study notes Reasoning in PDF only on Docsity!
Math 430: Notes on cardinality Friday 3/
Here is a summary of some of the extra topics we covered between the Midterm and Spring Break.
Definition 0.1 (4.18 in ML). A set A is countable if either it is empty or there is a surjective map g : N → A.
Wouldn’t it be nice if we could develop a notion of “size” for sets, where countable sets end up having the “same size” as N or less? The following develops a notion of “size” for a set A that in actuality is the class of all sets B that can be put into 1-1 correspondence with A. This is our definition of card(A), the “cardinality of A.”
- Equivalence relations For the following, understand a class to be a collection of sets that may not itself be a set.
Definition 1.1. Fix a set or class, X. A relation E ⊆ X × X is an equivalence relation on X iff it has the following properties
(1) (reflexivity) For all x ∈ X, E(x, x) (2) (symmetry) For all x, y ∈ X, E(x, y) ⇒ E(y, x) (3) (transitivity) For all x, y, z ∈ X, if E(x, y) and E(y, z), then E(x, z)
Definition 1.2. Let E be an equivalence relation on X. For a ∈ X we define the equivalence class of a (with respect to E) to be
[a]/E := {b ∈ X : E(a, b)}
We define an equivalence class (with respect to E) to be [c]/E for some c ∈ X.
By properties (1)-(3) in Def 1.1, two equivalence classes [a]/E, [c]/E are either completely disjoint, or identical. (Exercise) Moreover, every element of X is in some equivalence class. (Exercise) Therefore X is the disjoint union of equivalence classes with respect to E. We say that E partitions X, since it gives a decomposition of X into disjoint pieces.
- Cardinality
Definition 2.1. A function f : X → Y is a
- injection if it is 1-
- surjection if it is onto Y 1
- bijection if it is both 1-1 and onto Y
Definition 2.2. (1) Let Set be the class of all sets. (2) Define the equivalence relation E 0 ⊆ Set × Set as follows: for two sets A, B,
E 0 (A, B) if and only if there is a bijection f : A → B
Definition 2.3. Given a set A we define the cardinality of A to be
card(A) := [A]/E 0
Remark 2.4. For every set A, A ∈ card(A).
Definition 2.5. Say that card(A) ≤ card(B) if there is an surjection g : B → A.
The above definition is well-defined. (Exercise)
Proposition 2.6. (AC) There is a surjection h : B → A iff there is an injection g : A → B iff card(A) ≤ card(B).
Proof. (Exercise)
Definition 2.7. We say that a set A is countable if either A = ∅ or card(A) ≤ card(N)
The above definition is equivalent to Definition 4.18. (Exercise)
To extend our notion of “same size” from finite sets to infinite sets, we use the notion of bijective correspondence in Def 2.2: A and B are the “same size” if E 0 (A, B). Roughly, we understand that a set A is countable if it has “size” not greater than the “size” of N. Using Prop 2.6, countable sets may be put into 1-1 correspondence with a subset of N.
- Applications
Definition 3.1 (4.22 in ML). If M = (M, I) is an LA-structure, we say M is countable to indicate that M is a countable set.
Let A 0 = {<} where < is a binary relation symbol.
Definition 3.2. An LA′ -structure M = (M, I) is a dense linear order without endpoints if it satisfies the following LA-sentences (call the set consisting of the following sentences T 0 ):
(1) (∀x(¬(x < x)) ) (2) (∀x(∀y ((¬x =ˆy) → (x < y ∨ y < x)) )) (3) (∀x(∀y(∀z ((x < y ∧ y < z) → x < z) ))) (4) (¬(∃x(∀y (x =ˆy ∨ x < y) )))
