Sets
A set is a collection of distinct objects, called elements (or members). They are typically denoted by a capital letter (such as $A, B, C,$ etc.), and their elements can be written inside curly brackets. For example:
\[A=\{1, 2, 3, 4, e, 9.\overline9, 8086 \}\]Here, $1 \in A$ denotes that “$1$ is an element of $A$”, and $5 \notin A$ means that “$5$ is not an element of $A$”.
Types of Sets
| Name | Description | Example |
|---|---|---|
| Finite set | Contains a finite number of elements. | $B={1, 2, 3}$ |
| Infinite set | Contains an infinite number of elements | $\mathbb{N}= {0, 1, 2, 3, \dots}$ |
| Singleton set | Contains exactly one element. | $C={256}$ |
| Universal set | Contains all objects under consideration in a certain context. Denoted as $U$. | |
| Empty set | (Sometimes called the null set) Contains no elements, and is denoted as $\varnothing$ or ${}$ |
Subsets
A set $A$ is considered a subset of $B$ if every element in $A$ is also an element of $B$. In which case:
\[A \subseteq B\]which means: “$A$ is a subset of or equal to $B$”.
If $A$ is a subset of $B$, but $A \neq B$, then $A$ is considered a proper subset of $B$, shown as:
\[A \subset B\]The empty set ($\varnothing$) is a subset of every set, including singletons, since it has no elements.
Standard number sets
There are a range of standard number sets which the reader should be aware of:
| Symbol | Set name | Definition |
|---|---|---|
| $\mathbb{N}$ | Natural numbers | ${0, 1, 2, 3, \dots}$ |
| $\mathbb{Z}$ | Integers | ${\dots, -2, -1, 0, 1, 2, \dots}$ |
| $\mathbb{Q}$ | Rational numbers | $\tfrac{p}{q}$, where $p, q \in \mathbb{Z}, q \neq 0$ |
| $\mathbb{R}$ | Real numbers | All $\mathbb{Q}$, as well as irrationals such as $\pi, e, \sqrt{2}$ |
| $\mathbb{C}$ | Complex numbers | All numbers in the form $a+bi$ where $a, b \in \mathbb{R}, i^2 = -1$ |
As shown by Figure $5.1$, the sets are nested in the following way:
\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\]
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Note: The set of natural numbers ($\mathbb{N}$) is sometimes defined to exclude $0$. In this book, we state that $0 \in \mathbb{N}$ for consistency. Complex numbers are introduced fully in Chapter [[17. Complex Numbers]].