# Real Analysis

Some Preliminaries

# Sets

A *set* is a collection of objects. These objects are referred to as the *elements* of the set. For example, \(S = \{1, \delta, \circ, \{2, \gamma\}\}\). Note that \(|S| = 4\).

Another notation for writing down a set is \(S = \{x : P(x) \text{ is true}\}\), which means “\(S\) is a set of \(x\) such that \(P(x)\) is true”. \(P\) is a *property* and if it is true for \(x\), it will put \(x\) in \(S\). For example, \(S = \{x : x < 2\}\) means \(S\) is a set of \(x\) such that \(x\) is less than \(2\).

\(x \in S\) means \(x\) is in \(S\). \(x \notin S\) means \(x\) is not in \(S\). \(\emptyset\) is the *empty set*.

\(A \subset B\) means \(A\) is a *subset* of \(B\), or \(B\) *contains* \(A\), which means if \(x \in A\) then \(x \in B\) (or \(x \in A \implies x \in B\)).

If \(A \subset B\) and \(B \not\subset A\), then set \(A\) is a *proper subset* of \(B\).

If \(A \subset B\) and \(B \subset A\), then \(A = B\), else \(A \neq B\).

## Operations on Sets

Union: \(A \cup B = \{x : x \in A \text{ or } x \in B\}\).

Intersection: \(A \cap B = \{x : x \in A \text{ and } x \in B\}\).

Complement: \(A^c = \{x : x \notin A\}\).

Minus: \(A \setminus B = \{x : x \in A \text{ and } x \notin B\}\).

Product: \(A \times B = \{(x, y) : x \in A \text{ and } y \in B\}\).

# Relations

A *(binary) relation* \(R\) is a subset of \(A \times B\). If \((a, b) \in R\), then we can write \(aRb\).

For example, \(A\), “is an ancestor of”, is a relation on \(P \times P\), where \(P\) is a set of people. Another example is \(\lt\), “less than”, is a relation on \(\mathbb{Z} \times \mathbb{Z}\).

## Equivalent Relations

An *equivalent relation* \(R\) on \(S\), often denoted as \(\sim\), \(\approx\), \(\cong\), etc, is a relation on \(S \times S\) such that it is

- reflexive, i.e. \(aRa\) is true,
- symmetric, i.e. \(aRb \implies bRa\), and
- transitive, i.e. if \(aRb\) and \(bRc\) then \(aRc\).

## Functions

A *function* \(F\) from \(A\) to \(B\), denoted \(F : A \rightarrow B\), is a relation such that if \(aFb\) and \(aFb'\) then \(b = b'\). Informally, \(F\) assigns each \(a \in A\) to a single \(b \in B\). \(A\) is the *domain* of \(F\). \(B\) is the *codomain* of \(F\). The *range* of \(F\) is a subset of \(B\), given by \(\{y \in B : y = F(x) \text{ for some } x \in A\}\). We also write \(F(a) = b\).