# Set **A mathematical model for a collection of distinct objects.** The objects in a set are known as *elements* and may be mathematical objects of any kind, including other sets. The operations that act on sets are known as [[Set Algebra#Set operations|set operations]] whose behaviours are defined in [[set algebra]]. The *size* of a set is known as its [[cardinality]]. A set $A$ is a [[subset]] of set $B$ if all elements of $A$ are also elements of $B$. The [[power set]] of a set $A$ is the set of all subsets of $A$. ## Notation The elements of a set may be defined *explicitly* or *descriptively*. > [!Set membership] > The symbol for *set membership* is $\in$. > > If an element $x$ is in the set $S$, then > $x\in S$ > If an element $x$ is *not* in the set $S$, then > $x\notin S$ > [!Empty set] > The *empty set* is the set that contains *no members*. It is denoted variously as > $\varnothing, \emptyset, \{\}$ > The [[cardinality]] of the empty set is $0$. ^5f880b > [!Universal set or universe] > The *universal set*[^1] or the *universe*, $\mathcal{U}$, is the set containing *all elements* of *related sets* or all elements related to a particular situation. > > The universal set may also be called the *domain of discourse*, $\mathbb{D}$, particularly in the context of mathematical logic. ^1add85 ### Roster notation In *roster notation*, a set is defined by the elements which are within it. Sets are considered in this notation to be *unordered* collections of *distinct* objects. The set $S$ of elements $1$, $2$, and $3$ can be written as $S=\{1,2,3\}$ By considering sets as *unordered* collections, $S$ can also be written as $S=\{3,1,2\}$ As the elements of a set are considered *distinct*, repetition is ignored. $S=\{1,2,2,3,3,3\}=\{1,2,3\}$ ### Set-builder notation In *set-builder notation*, a set is defined as containing elements from a *larger set* with imposed conditions. A colon $:$ or vertical bar $|$ is can be interpreted as "such that". For example, a set $F$ of all numbers $n$ such that $n$ is an integer between $0$ to $4$ inclusive can be written as $F=\{n\mid n\in\mathbb{Z}, 0\le n\le 4\}=\{0,1,2,3,4\}$ ## Specific numeric sets These numeric sets are specific sets of numbers which are commonly referred to in various fields of mathematics. - $\mathbb{N}=\{0,1,2,3,4,\dots\}$ - *natural numbers*[^2] - $\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}$ - *integers* - $\mathbb{Q}=\{\frac{a}{b}\mid a,b\in\mathbb{Z},b\notin 0\}$ - *rational numbers* - $\mathbb{R}$ - *real numbers*, including rational and irrational numbers which include algebraic numbers and transcendental numbers. - $\mathbb{C}=\{a+bi\mid a,b\in\mathbb{R}\}$ - [[Complex Number|complex numbers]] [^1]: Sometimes the universal set is considered to be the set that contains *all objects including itself*, which does not exist in set theory as usually formulated. [^2]: It is debated whether $0$ should be considered a natural number.