Sets:Definitions

Definitions

A set is any collection of objects. The objects in a set are called elements. For example, $\mathbb {N} =\{1,2,3,...\}$ is the set of all natural numbers, and any positive integer is an element of this set. Sets do not necessarily have to contain mathematical elements. For example, we can say $P=$ {"red", "orange", "yellow", "green", "blue", "indigo", "violet"} is a set that contains the standard rainbow colors as its elements. To denote that an element $x$ is in a set $S$, we write $x\in S$. This is read as "$x$ is an element of $S$". If $x$ is NOT in the set $S$, we write $x\not \in S$. For example, looking at our set of colors $P$, we can see that "green" $\in P$ and "grey" $\not \in P$. If we consider the set of natural numbers, $5\in \mathbb {N}$, while $-7.5\not \in \mathbb {N}$.

Some important sets in mathematics:

• The empty set: $\emptyset$, which contains no elements. If a set $S$ is empty, we write $S=\emptyset$ or $S=\{\}$.
• The set of natural numbers: $\mathbb {N} =\{1,2,3,...\}$, which contains all positive integers (which numbers that can be written without decimals or fractions)
• The set of integers: $\mathbb {Z} =\{...,-2,-1,0,1,2,...\}$
• The set of rational numbers: $\mathbb {Q}$, which contains all numbers that can be written as ${\frac {p}{q}}$, where p and q are integers and $q\neq 0$. For example, ${\frac {8}{9}}$, ${\frac {-3}{1}}$, ${\frac {555}{2}}$, etc. Note that this set contains all integers.
• The set of real numbers: $\mathbb {R}$, which is all numbers in the interval $(-\infty ,\infty )$. This includes both rational numbers and irrational numbers (${\sqrt {2}}$, $\pi$, $1.01001000100001...$, and any other numbers that can't be expressed as a fraction of integers).

The set of irrationals doesn't have an official symbol, but it is typically stated as $\mathbb {R}$ without $\mathbb {Q}$ (that is, the real numbers without the rational numbers), denoted as $\mathbb {R} \backslash \mathbb {Q}$.

It is not always practical to list out the elements of a set. So, we can instead describe all of the elements in the set with specific criteria. For example, we could write the set of all even integers as $S=\{x:x=2k,k\in \mathbb {Z} \}$. That is, S contains all elements x such that x = 2k for some integer k. In this notation, we list our criteria for an element of set S after the colon within the braces. More examples of this notation:

• The rational numbers: $\mathbb {Q} =\{x:x=p/q,p\in \mathbb {Z} ,q\in \mathbb {Z} ,q\neq 0\}$.
• An open bounded interval: $(a,b)=\{x:a<x<b\}$
• A closed bounded interval: $[a,b]=\{x:a\leq x\leq b\}$
• The set of perfect squares: $S=\{x:x=k^{2},k\in \mathbb {Z} \}$

More definitions and notations for sets (let $A$ and $B$ be nonempty sets):

• If every element of $A$ is also an element of $B$, then $A$ is a subset of $B$, and we write $A\subseteq B$.
• If $A$ is not a subset of $B$ (that is, there is at least one element in $A$ that is not in $B$), then we write $A\not \subseteq B$.
• If $A\subseteq B$ and $B\subseteq A$, then $A=B$ (that is, $A=B$ iff $A$ and $B$ contain the exact same elements, and neither set contains an element that isn't in the other).
• If $A$ is a subset of $B$, and there is at least one element $b\in B$ such that $b\not \in A$ (that is, $A\subseteq B$, but $A\neq B$), then $A$ is a proper subset of B, and we write $A\subset B$. If $A$ is not a proper subset of $B$, we write that $A\not \subset B$.
• The power set of $A$, denoted as ${\mathcal {P}}(A)$, is the set of all subsets of $A$ (that is, ${\mathcal {P}}(A)=\{X:X\subseteq A\}$). For example, if $A=\{1,2,3\}$, then ${\mathcal {P}}(A)=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$. Note that if $A$ contains $n$ elements, then ${\mathcal {P}}(A)$ contains $2^{n}$ elements. ${\mathcal {P}}(A)$ will always contain $\emptyset$ and $A$ itself.

Resources

• Course Textbook, pages 93-100