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0x000 Set Theory

This first note sets up the formal notations of sets, logic and number systems used in all notes.

1. History

The naive set theory, unlike the axiomatic set theories, is using informal logics based on natural languages. The naive set theory was first created by Georg Cantor at the end of 19th century and developed by Gottlob Frege.

However, Bertrand Russel found the following paradox during his work on Principia Mathematica

\[A = \{X | X \not\in X \}\]

from which contradictory conclusion can be drawn when deciding whether \(A \in A\).

The paradox comes from the following axiom assumption.

Axiom (Universal specification, Axiom of comprehension): Suppose for every object \(x\), we have a property \(P(x)\) pertaining to \(x\). Then there exists a set \(\{ x | P(X) \text{ is true } \}\)

This is the one of examples of self-reference, which has many interesting applications in mathematics and computer science. For example, Gödel, Escher, Bach has many examples about this.

Frege respond to the letter from Russel as follows, sad..

Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished Gottlob Frege

2. Zermelo–Fraenkel

Axiom (Axiom of extensionality, equality of set). Two sets \(A\) and \(B\) are equal, \(A=B\), iff every element of \(A\) is an element of \(B\) and vice versa.

Axiom (Axiom of pairing) If \(a\) is an object, then there exists a set \(\{ a \}\) (singleton). If \(a\) and \(b\) are objects, then there exists a set \(\{ a, b \}\) (pair)

Axiom (Axiom of regularity, axiom of foundation) If \(A\) is a non-empty set, then there is at least one element \(x\) of \(A\) which is either not a set, or is disjoint from \(A\).

For any set \(A\), applying pairing axiom to create a singleton set \(\{ A \}\). Then applying the regularity shows that \(A \cap \{ A \} = \emptyset\), this indicates that \(A \notin A\) (i.e. no set is an element of itself)

This rules out the existence of the universal set (the set contains all objects including itself)

Axiom (Pairwise union): Given any two sets \(A, B\), there exists an union set \(A \cup B\)

Note that Union operation is commutative and associative

Definition (Subsets) Let \(A,B\) be sets. We say that \(A\) is a subset of \(B\), denoted \(A \subseteq B\) iff every element of \(A\) is also an element of \(B\).

Axiom (Axiom of specification, axiom of separation) Let \(A\) be a set, and for each \(x \in A\), let \(P(x)\) be a property pertaining to \(x\) (either true or false). Then there exists a set, called \(\{ x \in A | P(X) \}\)

intersection and difference

Axiom of sepcification can be used to construct intersection and difference:

The intersection \(S_1 \cap S_2\) of two sets is defined to be the set

\[S_1 \cap S_2 = \{ x \in S_1 | x \in S_2 \}\]

Given two sets \(A\) and \(B\), we define the set \(A-B\) or \(A \setminus B\)

\[A \setminus B := \{ x \in A | x \not\in B \}\]

Proposition (Sets form a boolean algebra) Let \(A,B,C\) be sets, and let \(X\) be a set containing \(A,B,C\) as subsets

  • (minimum element) \(A \cup \emptyset = A\) and \(\emptyset \cap A = \emptyset\)
  • (maximum element) \(A \cup X = X\) and \(A \cap X = A\)
  • (identity) \(A \cap A = A, A \cup A = A\)
  • (commutativity) \(A \cup B = B \cup A\) and \(A \cap B = B \cap A\)
  • (associativity) \((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)
  • (distributivity) \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C), A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)
  • (Partition) \(A \cup (X \setminus A) = X, A \cap (X \setminus A) = \emptyset\)
  • (De morgan laws) \(X \setminus (A \cup B) = (X \setminus A) \cap (X \setminus B), X \setminus (A \cap B) = (X \setminus A) \cup (X \setminus B)\)

Axiom/definition so far only create sets with finite members

Axiom (axiom of infinity) There exists a set \(\mathcal{N}\), whose elements are called natural numbers such that Peano axioms hold

Axiom (Replacement) Let \(A\) be a set. For ay object \(x \in A\), and any object y, suppose we have a statement \(P(x,y)\) pertaining to \(x\) and \(y\), such that for each \(x \in A\) there is at most one \(y\) for which \(P(x,y)\) is true. Then there exists a set \(\{ y | P(x,y) \text{ is true for some } x \in A \}\)

This can be used to apply function to element of a set to create another set

3. Function

There are two ways to formally define the function, the common way is to simply define the domain, range, and how one generates the output \(f(x)\) from each input, this is known as an explicit definition of a function, more or less the one proposed by Peter Lejeune Dirichlet in the 1830s when studying the convergence of Fourier Series.

The other case is to efine a function by specifying what property \(P(x,y)\) links the input \(x\) with the output \(f(x)\); this is an implicit definition of a function.

Definition (Functions) Let \(X,Y\) be sets, and let \(P(x,y)\) be a property pertaining to an object \(x \in X\) and an object \(y \in Y\), such that for every \(x \in X\) there is exactly one \(y \in Y\) for which \(P(x,y)\) is true. Then we define the function \(f: X \rightarrow Y\) defined by \(P\) to be the object which, given any input \(x \in X\), assigns an output \(f(x) \in Y\) for which \(P(x, f(x))\) is true.

Definition (Equality of functions) Two functions \(f: X \rightarrow Y, g: X \rightarrow Y\) with the same domain and range are said to be equal \(f=g\) iff \(f(x) = g(x)\) for all \(x \in X\)

Definition (Composition) Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be two functions, such that the range of \(f\) is the same set as the domain of \(g\). We then define the composition \(g \circ f: X \rightarrow Z\) of the two functions \(g\) and \(f\) to be the function defined explicitly by the formula

\[(g \circ f)(x) := g(f(x))\]

Lemma (Composition is associative) Let \(f: Z \rightarrow W, g: Y \rightarrow Z\) and \(h: X \rightarrow Y\) be functions. Then \(f \circ (g \circ h) = (f \circ g) \circ h\)

Definition (injective) A function \(f\) is one-to-one (or injective) if different elements map to different elements

\[x \neq x' \implies f(x) \neq f(x') \]

Definition (surjective) A function \(f\) is onto (or surjective) if every element in \(Y\) comes from applying \(f\) to some element in \(X\): for every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\)

Definition (bijective) A function is called bijective or invertible if it is both injective and surjective

3.1. Images and Inverse Images

Definition (Images of Sets) If \(f: X \rightarrow Y\) is a function from \(X\) to \(Y\), and \(S\) is a set in \(X\), we define \(f(S)\) to be the set

\[f(S) := \{ f(x) | x \in S \}\]

Definition (Inverse images) If \(U\) is a subset of \(Y\), we define the set \(f^{-1}(U)\) to be the set

\[f^{-1}(U) := \{ x \in X | f(x) \in U \}\]

Lemma (algebra of inverse images) Suppose \(f: X \to Y\) is a function, then

\[f^{-1}(Y \setminus A) = X \setminus f^{-1}(A)\]
\[f^{-1}(\cup_{A \in \mathcal{A}} A) = \cup_{A \in \mathcal{A}} f^{-1}(A)\]
\[f^{-1}(\cap_{A \in \mathcal{A}} A) = \cap_{A \in \mathcal{A}} f^{-1}(A)\]

Axiom (Power set axiom) Let \(X,Y\) be sets. Then there exists a set, denoted \(Y^{X}\), which consists of all the functions from \(X\) to \(Y\)

Remark: this is to basically create larger set such as \(\mathbf{R}\)

Lemma (existence of power set) Let \(X\) be a set, then following set exists

\[\{ Y | Y \subseteq X \}\]

Axiom (Union) Let \(A\) be a set, all of whose elements are themselves sets. Then there exists a set \(\cup A\) whose elements are precisely those objects which are elements of the elements of \(A\).

4. Cartesian Products

Definition (Ordered Pair) If \(x\) and \(y\) are any objects, we define the ordered pair \((x,y)\) to be a new object, consisting of \(x\) as its first component and \(y\) as its second component. Two ordered pairs \((x,y)\) and \((x', y')\) are considered equal iff both their components match

Remark: pair can be implemented with set as \((x,y) = \{ \{ x \}, \{ x, y \} \}\)

Definition (Cartesian Product) If \(X\) and \(Y\) are sets, then we define the Cartesian product $X \times Y $ to be the collection of ordered pairs

\[X \times Y = \{ (x,y) | x \in X, y \in Y \}\]

Definition (Ordered n-tuple and n-fold Cartesian product) Let \(n\) be a natural number. An ordered n-tuple \((x_i)_{1 \leq i \leq n}\) (also denoted \((x_1, ..., x_n)\) is a collection of objects \(x_i\). If \((X_i)_{1 \leq i \leq n}\) is an ordered n-tuple of sets, we define their Cartesian product \(\prod_{1 \leq i \leq n} X_i\) (also denoted \(X_1 \times ... \times X_n\)) by

\[\prod_{1 \leq i \leq n} := \{ (x_i)_{1 \leq i \leq n} | x_i \in X_i \text{ for all } 1 \leq i \leq n \}\]

Definition (Finite choice) Let \(n \geq 1\) be a natural number and for each natural number $ 1 \leq i \leq n$, let \(X_i\) be a non-empty set. Then there exists an n-tuple \((x_i)_{1 \leq i \leq n}\) such that \(x_i \in X_i\) for all \(1 \leq i \leq n\)

Remark: infinite number of choices requires the axiom of choice

6. Relations

Definition (relation) A relation on a set \(A\) is a subset \(C\) of the Cartesian product \(A \times A\)

The following two relations are important especially.

6.1. Equivalence Relation

Definition (equivalence relation) An equivalence relation on a set is a relation \(C\) having the following properties:

  • (reflexivity) \((\forall x \in A) xCx\)
  • (symmetry) \(xCy \implies yCx\)
  • (transitivity) \(xCy, yCz \implies xCz\)

Definition (equivalence class) Given an equivalence relation \(\sim\) on a set \(A\) and \(x \in A\), we define a subset \(E\) of \(A\), called the equivalence class defined by \(x\) by

\[E = \{ y | y \sim x \}\]

Lemma Two equivalence class \(E, E'\) are either disjoint or equal

Definition (partition) A partition of \(A\) is a collection of disjoint nonempty subsets of \(A\) whose union is all of \(A\)

6.2. Order Relation

Definition (order relation) A relation \(C\) on a set \(A\) is called an order relation if it has the following properties:

  • (comparability) \(\forall x, y \in A, xCy \text{ or } yCx\)
  • (nonreflexivity) \(xCx\) does not hold for all \(x\)
  • (transitivity) \(xCy, yCz \implies xCz\)

The order relation gives the defintion of open interval \((a,b)\)

\[(a, b) := \{ x | a < x < b \}\]

If \((a,b)\) is empty, we call \(a\) is the immediate predecessor of \(b\) and \(b\) is the immediate successor of \(a\)

A new order can be constructed by taking Cartesian product

Definition (dictionary order) Suppose \(A, B\) are sets with order \(<_A, <_B\), then we can define order relation \(<\) on \(A \times B\) by

\[a_1 \times b_1 < a_2 \times b_2\]

if \(a_1 < a_2\) or \(a_1 = a_2\) and \(b_1 < b_2\).

5. Cardinality

To compare the size of different sets, Cantor's idea was to attempt to put the sets into a 1-1 correspondence with each other.

Definition (equal cardinality) \(X\) and \(Y\) have equal cardinality iff there exists a bijection \(f: X \rightarrow Y\). In this case, we write

\[X \sim Y\]

The notion of having equal cardinality is an equivalence relation (i.e: reflective, symmetric, transitive)

Definition (cardinality) Let \(n\) be a natural number. A set \(X\) is said to have cardinality \(n\) iff it has equal cardinality with ${ i \in N | i < n } $. We also say that \(X\) has \(n\) elements iff it has cardinality of \(n\)

Proposition (uniqueness of cardinality) Let \(X\) be a set of some cardinality \(n\). Then \(X\) cannot have any other cardinality.

Lemma Suppose that \(n \geq 1\) and \(X\) has cardinality \(n\). Then \(X\) is non-empty, and if \(x\) is any element of \(X\), then the set \(X - \{ x \}\) has cardinality \(n-1\).

Definition (finite, infinite) A set is infinite iff it has cardinality \(n\) for some natural number \(n\); otherwise, the set is called infinite. If \(X\) is a finite set, we use \(\#(X)\) to denote the cardinality of \(X\)

In the infinite sets, it has more than 1 level of cardinality.

Definition (countable, uncountable) A set \(A\) is countable if \(\mathbb{N} \sim A\), an infinite set that is not countable is called an uncountable set. (e.g: N, Z, Q are countable and R is uncountable)

Cantor's Diagonalization

The open interval \((0,1)\) is uncountable. This can be shown by expanding each real number into a decimal representation.

7. Reference

[1] Hofstadter, Douglas R. Gödel, escher, bach: ein endloses geflochtenes band. Klett-Cotta, 2006.

[2] Tao, Terence. Analysis. Vol. 1. Hindustan Book Agency, 2006.