The Mathematics of ECE

Real Analysis - Lecture 1

Monday, September 19, 2022

Logistics

  • Housekeeping
  • Today’s class: Real Analysis
    • Key concepts: Rationals, reals, countability, open and closed sets, basic topology, interior and closure.
  • Next workshop: Wednesday September 21, 2022

Real Numbers

  • A rational number \(x \in \bbQ\) is one such that there exists two integers \(p, q \in \bbZ\) for which \(x = \frac{p}{q}\).

  • Drill: Is \(\sqrt{2}\) a rational number?
    • So not every quantity can be described with a rational number…
  • Drill: Between any two rational numbers, does there exist another rational number?
    • So the idea of a “next” rational number is ill-defined…

  • An irrational number \(x \in \bbR \setminus \bbQ\) is one which is not rational.

Topology (Preliminaries)

  • Let \(\calA, \calB, \calC\) be sets. The union is denoted \(\cup\) and the intersection \(\cap\) with definitions \[\begin{align*} \calA \cup \calB &= \{ x | ~ x \in \calA \lor x \in \calB \} \\ \calA \cap \calB &= \{ x | ~ x \in \calA \land x \in \calB \} \end{align*}\] which obey the following properties
    • \(\calA \cup \calB = \calB \cup \calA\) and \(\calA \cap \calB = \calB \cap \calA\) (commutativity)
    • \((\calA \cup \calB) \cup \calC = \calA \cup (\calB \cup \calC)\) and \((\calA \cap \calB) \cap \calC = \calA \cap (\calB \cap \calC)\) (associativity)
    • \(\calA \cup (\calB \cap \calC) = (\calA \cup \calB) \cap (\calA \cup \calC)\) and \(\calA \cap (\calB \cup \calC) = (\calA \cap \calB) \cup (\calA \cap \calC)\) (distributivity)

  • Let \(\calA\) be some set. If for some \(n \in \bbN\) there exists a one-to-one mapping between elements of \(\calA\) and \(1,2,\dots,n\), then \(\calA\) is finite; otherwise, \(\calA\) is infinite.

  • Let \(\calA\) be some infinite set. If there exists a one-to-one mapping between elements of \(\calA\) and \(\bbN\), then \(\calA\) is countable; otherwise, \(\calA\) is uncountable.

  • Drill: Is \(\bbN\) (the naturals) finite? Countable?
  • Drill: Is \(\bbQ\) (the rationals) finite? Countable?
  • Drill: Is \(\bbR\) (the reals) finite? Countable?

Topology

  • For any vector \(x \in \bbR^n\) and real number \(r \in \bbR\), we define the (open) ball \(\calB_r(x)\) centered at \(x\) with radius \(r > 0\) as \[ \calB_r(x) = \{ y \in \bbR^n | ~ || x - y || < r \} \] We call a subset \(\calA \subseteq \bbR^n\) an open set if it is some union (finite or infinite) of balls. The empty set is considered open.

  • Drill: Is \(\bbR\) an open set?
  • Drill: Prove that for any \(x_a, x_b \in \bbR^n, r_a, r_b \in \bbR\) the intersection \(\calB_{r_a}(x_a) \cap \calB_{r_b}(x_b)\) is an open set.
  • Drill: Prove that the intersection of any two open sets \(\calA, \calB\) is an open set.
    • Hint: Use the previous result and some set algebra.
  • Drill: Is the intersection of any collection of open sets also an open set?

  • A set \(\calA\) is closed if its complement is an open set.

  • Drill: Is \(\bbR\) a closed set?
  • Drill: Prove that the intersection of some collection (finite or infinite) of closed sets is closed.
  • Drill: Prove that the union of a finite collection of closed sets is closed.
  • Drill: Is every set either open or closed?

Topology

  • Let \(\calA\) be an arbitrary set. A point \(x \in \calA\) is an interior point of \(\calA\) if there exists some \(r > 0\) such that \(\calB_r(x) \subseteq \calA\). The collection of all interior points is called the interior of \(\calA\) and denoted \(\mathring{\calA}\).

  • Let \(\calA\) be an arbitrary set. A point \(x \in \calA\) is a closure point of \(\calA\) if for all \(r > 0\) there exists some \(y \in \calA\) such that \(y \in \calB_r(x)\). The collection of all closure points is called the closure of \(\calA\) and denoted \(\bar{\calA}\).

  • A neighborhood \(\calN_x\) of a point \(x\) is a set such that \(x\) is an interior point of \(\calN_x\).

  • Drill: Prove that all neighborhoods of a point \(x\) contain a subset which is an open neighborhood of \(x\).