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 \(F(a) = b\).