Set notation

Notation for some commonly used sets.

• $\mathbb{Z}$, the integers. $\{\ldots, -2, -1, 0, 1, 2 \ldots\}$
• $\mathbb{Z}_{\geq 0}$, the nonnegative integers. $\{0, 1, 2, \ldots\}$
• $\mathbb{N}$, the positive integers (also known as the natural numbers), $\{1, 2, 3 \ldots\}$
• $\mathbb{R}$, the real numbers. Not possible to list.

Set and element relations.

• $\in$, element of. If $x$ is a member of (or an element of) a set $\mathcal{S}$, we write $x \in \mathcal{S}$.
• $\subset$, subset of. If all of the elements of a set $A$ are also elements of a set $\mathcal{S}$, we write $A \subset \mathcal{S}$.
• $:$, such that. This symbol is used when we want to construct a subset of elements that satisfy a specific condition. For example, if we wish to specify the set of all natural numbers less than 12, we write $A = \{x \in \mathbb{N} \, : \, x < 12\}$.

Focusing on the real numbers, an interval $I$ is a subset of the real numbers that contains all real numbers between two endpoints $a < b$ and no numbers less than $a$ or greater than $b$.

• We say that an interval is open if it does not contain its endpoints, i.e., if it has the form $I = \{x \in \mathbb{R} \, : \, a < x < b\}$. We write $I = (a,b)$.
• We say that an interval is closed if it does contain its endpoints, i.e., if it has the form $I = \{x \in \mathbb{R} \, : \, a \leq x \leq b\}$. We write $I = [a,b]$.
• An interval can be half-open if contains just one of its endpoints. Either $I = (a,b]$ or $I = [a,b)$.
• Half-line. If we construct a set that contains all real numbers greater than an endpoint, we call this a half-line and write the intervals as if infinity, $\infty$, is a number, but it cannot be contained in the interval:
  • $(a, \infty) = \{x \in \mathbb{R} \, : \, x > a\}$,
  • $[a,\infty) = \{x \in \mathbb{R} \, : \, x \geq a\}$,
  • $(\infty,a) = \{x \in \mathbb{R} \, : \, x < a\}$,
  • $(\infty,a] = \{x \in \mathbb{R} \, : \, x \leq a\}$.

We will make a critical distinction between two types of sets: discrete and continuous. When we visualize discrete sets, we will mark them as distinct dots on the number line. When we visualize continuous sets (which are just intervals for the real line), we will mark them by shading the number line in the appropriate region and marking the endpoints with either open or filled in circles to indicate whether they are contained in the set.

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Discrete and Continuous Sets (of Real Numbers)

A set of real numbers $\mathcal{S} \subset \mathbb{R}$ is called discrete if for every element $x \in \mathcal{S}$, there exists an open set $A \subset \mathbb{R}$ that contains $x$ and no other members of $\mathcal{S}$.

A set of real numbers $\mathcal{S} \subset \mathbb{R}$ will be called continuous if for any two elements $x, y \in \mathcal{S},$ if $t \in \mathbb{R}$ satisfies $x < t < y$, then $t \in \mathcal{S}$.

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Typically the term “continuous” is not applied to sets. It is usually reserved for functions. This definition will be helpful for us a little later when we define discrete and continuous random variables.

Let $A$ and $B$ be two subsets of a common set $\mathbb{\Omega}$.

• Intersection. $A \cap B \coloneqq \{x \in \mathcal{S} \, : \, x \in A \text{ and } x \in B\}$.