Notes on Cardinality and Equivalence Relations, Study notes of Reasoning

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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.”

1. 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.

2. 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.

3. 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) )))

(5) (¬(∃x(∀y (x =ˆy ∨ y < x) ))) (6) (∀x(∀y (x < y → (∃z (x < z ∧ z < y))) )) We may define a new relation ≤ as follows: M  (∀x(∀y (x ≤ y ↔ (x =ˆy ∨ x < y)) ))

Theorem 3.3 (4.24 in ML). Suppose that M = (M, <) is a countable dense total order. Then M and (Q, <) are isomorphic.

Definition 3.4. By an LA-theory we mean a set of sentences from LA.

Theorem 4.24 tells us that the theory T 0 is what we call countably categorical : any two countable structures satisfying the theory are isomorphic. Clearly two structures satisfying T 0 that are of different cardinality cannot be put into bijection with one another, thus, are not isomorphic. But if cardinality is the only impediment to isomorphism, in some sense, then this is a powerful property of the theory.

Definition 3.5. An LA-theory T is complete if for any LA-sentence ϕ, either T ϕ or T (¬ϕ).

As a consequence of Theorem 4.24 and some further work, the theory T 0 listed in Def 3. is complete. This is a special property of theories that we often look for. The exercise below gives one reason why.

Definition 3.6. For an LA-structure M, let Th(M) = {ϕ ∈ LA : M  ϕ}

Definition 3.7. For an LA-structure M and an LA-theory T , by

M  T

we mean M  ϕ for all ϕ ∈ T.

Exercise 3.8. If T is a complete LA-theory, then for any two LA-structures M, N

M  T, N  T ⇒ M ≡ N (Hint: it is enough to show that Th(M)=Th(N ).)