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:
  • $(a, \infty) = \{x \in \mathbb{R} \, : \, x > a\}$,
  • $[a,\infty) = \{x \in \mathbb{R} \, : \, x \geq a\}$,
  • $(\infty,a) = \{x \in \mathbb{R} \, : \, x < a\}$,
  • $(\infty,a] = \{x \in \mathbb{R} \, : \, x \leq a\}$.

A special class of sets are discrete sets. When we visualize discrete sets, we will mark them as distinct dots on the number line. This is in contrast to how we will visualize intervals, which will be marked 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 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 $\mathcal{S}$.

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

• Intersection. $A \cap B \coloneqq \{x \in \mathcal{S} \, : \, x \in A \text{ and } x \in B\}$.
• Union. $A \cup B \coloneqq \{x \in \mathcal{S} \, : \, x \in A \text{ or } x \in B\}$
• Complement within $\mathcal{S}$. $A^c$ or $A' \coloneqq \{x \in \mathcal{S} \, : \, x \notin A\}$.