3.1 Set theory

Set Definition

A set is a primitive notion. That is, it is a basic idea of human thought. A set is a collection of objects: numbers, images … anything, football players, words, colors … The basic set theory is simple and natural and is what we need for this course. Mathematical set theory is more complex and presents several paradoxes. The sets usually have a mother set such as:

  • \(\mathbb{N}=\{0,1,2,\ldots\}\)
  • \(\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}\)
  • \(\mathbb{Q}=\left\{\frac{p}{q}\quad\Big|\quad p,q\in \mathbb{Z} \mbox{ and } q \not= 0.\right\}\)
  • \(\mathbb{R}=\{\mbox{All points in a line.}\}\)
  • Alphabet = \(\{a,b,c,\ldots, A,B,C,\ldots\}.\)
  • Words = \(\{peace, war, love, probability,\ldots\}.\)

Each \(\omega\) object of \(\Omega\) is called the element of the set \(\Omega\) and we say that \(\omega\) belongs to \(\Omega\). We will denote it with \(\omega\in \Omega\). A set of one element, for example, \(\{1\}\) is called a singleton.

Let \(A\) be another set we will say that \(A\) is equal to \(B\) if all the \(A\) elements are in \(B\) and all the \(B\) elements are in \(A\). For example \(A=\{1,2,3\}\) is equal to \(B=\{3,1,2\}\). If \(A\) is another set such that if \(x\in A\) then \(x\in B\) we will say that \(A\) is a subset of or is contained in \(B\). We will denote it by \(A\subseteq B.\)

The set that has no elements is called an empty set and is denoted by the symbol \(\emptyset\). Given \(A\) a set obviously \(\emptyset\subseteq A.\)

Let’s take as a base set \(\Omega=\{1,2,3\}\)

\(\Omega\) is a set of cardinal 3, it is denoted by \(\#(\Omega)=3\) or by \(|\Omega|=3\). The \(\Omega\) set has \(2^3=8\) subsets:

  • the empty \(\emptyset\) and the elementaries \(\{1\},\{3\},\{3\}\)
  • the subsets of two elements: \(\{1,2\},\{1,3\},\{2,3\}\)
  • the total set of three elements \(\Omega=\{1,2,3\}.\)

3.1.1 Sample Spaces

Probability is the mathematical language we use to quantify uncertainty.

  • The sample space \(\Omega\) is the set of possible outcomes of an experiment.
  • Points \(\omega\in\Omega\) are called outcomes, realizations, or elements.
  • Subsets of \(\Omega\) are called events

Example: If we toss a coin twice then \(\Omega =\{ HH, HT, TH, TT\}\). The event that the first toss is heads is \(A=\{HH,HT\}\).

Example: Let \(\omega\) be the outcome of a measurement of some physical quantity, for example, temperature. Then \(\Omega=\mathbb{R}=(-\infty,\infty)\) (without considering that there is a lower bound, no harm in this). The event that the measurement is larger than 10 but less than or equal to 23 is \(A=(10,23]\).

Example: If we toss a coin forever, then the sample space is the infinite set \(\Omega = \{w=(w_1,w_2,w_3,\dots): w_i\in\{H,T\}\}\). Let \(E\) be the event that the first head appears on the third toss. Then

\[ E=\{(w_1,w_2,w_3,\dots):w_1=T,w_2=T,w_3=H,w_i\in\{H,T\}\;\; for\;\; i>3\}. \]