# 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

1. reflexive, i.e. $$aRa$$ is true,
2. symmetric, i.e. $$aRb \implies bRa$$, and
3. 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 $$F(a) = b$$.