Typically, an uppercase letter is used to denote a set. Sets don't necessarily have to hold numbers - they can hold virtually any element such as letters, variables and outcomes:

$$\begin{align*} C&=\{\text{h},\text{e},\text{l},\text{l},\text{o}\}\\ D&=\{d_1,d_2,d_3\}\\ E&=\{\text{heads},\text{tails}\}\\ \end{align*}$$

Since the ordering of the elements within a set does not matter, two sets are considered equivalent if they contain the same elements. For instance, the following two sets are equivalent:

$$\{1,2\}=\{2,1\}$$

Special sets

The following are special sets that serve as the backbone of mathematics:

the set $\mathbb{Z}=\{\cdots,-2,-1,0,1,2,\cdots\}$ is the set of all integers.
the set $\mathbb{N}=\{1,2,3,\cdots\}$ is the set of all natural numbers.
the set $\mathbb{Q}$ is the set of all rational numbers.
the set $\mathbb{R}$ is the set of all real numbers.
the set $\mathbb{C}$ is the set of all complex numbers.

To emphasize the fact that these are special sets, they are written in a font called blackboard bold - for instance, we write $\mathbb{R}$ instead of $R$.

Empty sets

A set that contains no elements is called an empty set, represented by the symbol $\emptyset$.

Universal sets

A universal set contains all the elements under consideration. For instance, suppose we roll a dice once - the universal set $S$ could represent all possible outcomes:

$$S=\{1,2,3,4,5,6\}$$

We will discuss more about universal sets in the section about Venn diagramslink later.

Set notations

Containment

To indicate that an element belongs to a set, we use the $\in$ symbol. For instance, consider the following set:

$$A=\{1,3,5\}$$

To state that $3$ is contained in this set $A$:

$$3\in{A}$$

In contrast, if an element does not belong to a set, then we use the $\notin$ symbol. For instance, to indicate that $4$ is not contained in our set $A$:

$$4\notin{A}$$

The containment symbol is extremely common in mathematics because we typically characterize the nature of a variable using $\in$. For instance, to indicate that a variable $x$ is an integer, we would write $x\in\mathbb{Z}$. In linear algebra, we denote a vector $\boldsymbol{v}$ in two-dimensional space as $\boldsymbol{v}\in\mathbb{R}^2$.

Subsets

We say that set $A$ is a subset of set $B$ if and only if all elements in set $A$ are also contained in set $B$. For instance, consider the following sets $A$ and $B$ that contain some numbers:

$$\begin{align*} A&=\{1,2,4\}\\ B&=\{1,2,3,4,10\}\\ \end{align*}$$

Here, $A$ is a subset of $B$ because all the numbers within $A$ also belong to $B$. We can mathematically express this as follows:

$$A\subset{B} \;\;\;\;\text{or}\;\;\;\; B\supset{A}$$

Note that $A\subset{B}$ is to be read as "$A$ is a subset of $B$". Equivalently, we can call $B$ the superset of $A$, that is, $B$ contains all elements of $A$.

As another example, consider the set of natural numbers $\mathbb{N}$ and integers $\mathbb{Z}$:

$$\mathbb{N}\subset\mathbb{Z}$$

This means that the set of natural numbers is a subset of the set of integers.

Intersection

The intersection of sets $A$ and $B$, which is denoted by $A\cap{B}$, is the set of all elements that are contained in both $A$ and $B$. For instance, consider the following set:

$$\begin{align*} A&=\{1,2,5\}\\ B&=\{1,2,3,9\}\\ \end{align*}$$

The numbers $1$ and $2$ both belong to sets $A$ and $B$. Therefore, the intersection of $A$ and $B$ is:

$$A\cap{B}=\{1,2\}$$

Notice how intersections are guaranteed to be subsets of both $A$ and $B$, that is:

$$(A\cap{B})\subset{A},\;\;\;\;\;\;\; (A\cap{B})\subset{B}$$

Disjoint sets

Two sets $A$ and $B$ are considered disjoint sets if there is no common element between them. For instance, consider the following sets:

$$\begin{align*} A&=\{1,2,4\}\\ B&=\{3,5,6,7\}\\ \end{align*}$$

Because there exists no element that belongs to both sets $A$ and $B$, we say that $A$ and $B$ are disjoint sets. Mathematically, disjoint sets are expressed as:

$$A\cap{B}=\emptyset$$

This means that the intersection of $A$ and $B$ is the empty set.

Union

The union of sets $A$ and $B$, which is denoted by $A\cup{B}$, is the combined set of elements in $A$ or $B$. For instance, consider the following sets:

$$\begin{align*} A&=\{1,2,4\}\\ B&=\{1,2,5\}\\ \end{align*}$$

The union of sets $A$ and $B$ is:

$$A\cup{B}=\{1,2,4,5\}$$

Notice how numbers $1$ and $2$ are contained in both $A$ and $B$, but they only appear once in the union. In other words, duplicates are removed in the resulting union.

Exclusion

The operation $A\setminus{B}$ returns a set of elements in $A$ that do not exist in $B$. For instance, consider the following sets:

$$\begin{align*} A&=\{1,2,4\}\\ B&=\{1,4,5\}\\ C&=\{1,2\}\\ \end{align*}$$

To remove elements in $A$ and $B$ that belong to $C$, use the $\setminus$ operation like so:

$$\begin{align*} A\setminus{C}&=\{4\}\\ B\setminus{C}&=\{4,5\}\\ \end{align*}$$

Cardinality

Cardinality of a set refers to the number of elements within a set. For instance, consider the following set:

$$A=\{1,2,4\}$$

The cardinality of $A$, denoted as $\vert{A}\vert$, is equal to 3.

NOTE

Since empty sets do not contain any elements, their cardinality is $0$, that is, $\vert{\emptyset}\vert=0$.

Set-builder notation

Up to now, we have only defined sets explicitly like $A=\{1,2,3\}$. However, we can also define sets by specifying using the set-builder notation. For instance, let's define a set that contains even integers:

$$\begin{equation}\label{eq:ufvKee1gfATYcRyK2an} A=\{x\in{\mathbb{Z}}\;|\;x\text{ is even}\} \end{equation}$$

This is read as "the set of all integers $x$ such that $x$ is even". The vertical line $|$, which means "such that", is sometimes written as $:$ instead. The set-builder notation is quite flexible - we can also equivalently write \eqref{eq:ufvKee1gfATYcRyK2an} as follows:

$$\begin{equation}\label{eq:dV2gcJtkk9ewMjW95AC} A=\{x\;|\;x\in{\mathbb{Z}}\text{ and }x\text{ is even}\} \end{equation}$$

As another example, let's define another set that holds natural numbers less than $8$:

$$A=\{x\in{\mathbb{N}}\;|\;x\lt8\}$$

Finally, here's another one:

$$A=\{(x,y)\;|\;x\in\mathbb{R}\text{ and }y=x^2\}$$

This set contains all the points on a parabola!

Intervals

Closed intervals

The endpoints are included in the interval. These intervals are expressed using square brackets:

$$[a,b]=\{x\in\mathbb{R}\;|\;a\le{x}\le{b}\}$$

Here's an example:

$$[3,8]=\{x\in\mathbb{R}\;|\;3\le{x}\le{8}\}$$

The ends points $3$ and $8$ are included in this interval.

Open intervals

The endpoints are not included in the interval. These intervals are expressed using parentheses:

$$(a,b)=\{x\in\mathbb{R}\;|\;a\lt{x}\lt{b}\}$$

The endpoints $a$ and $b$ are excluded from the interval.

Mixing open and closed intervals

We could also mix the notation of open and closed intervals:

$$\begin{align*} [a,b)&=\{x\in\mathbb{R}\;|\;a\le{x}\lt{b}\}\\ (a,b]&=\{x\in\mathbb{R}\;|\;a\lt{x}\le{b}\} \end{align*}$$

Denoting elements that belong to an interval

Suppose we wanted to mathematically state that a variable $x$ lies in the interval from $2$ to $6$ where both endpoints are included. We can use the containment symbol from earlier:

$$x\in[2,6]$$

This notation is extremely common in calculus!

Venn diagrams to visualize sets

Venn diagrams are a useful way to visualize the relationship between sets. Typically, we use circles to represent sets, but we can in fact use any shape we wish. Consider the following set:

$$A=\{1,2,3\}$$

We can represent this set using a Venn diagram like so:

We can think of the circle as the container of elements, just as the curly brackets $\{\cdots\}$ are in set notation.

Intersection

Recall that the intersection of sets $A$ and $B$, which is denoted by $A\cap{B}$, is the set of all elements that are contained in both $A$ and $B$. For example, consider the following two sets:

$$\begin{equation}\label{eq:B4V9tinT2Y5LHf7E2Q7} \begin{aligned} A&=\{1,2,3\}\\ B&=\{3,4,5\}\\ \end{aligned} \end{equation}$$

The intersection $A\cap{B}$ can be visualized in a Venn diagram like so:

Here, the shaded area represents the intersection of the two sets.

Union

Recall that the union of sets $A$ and $B$, which is denoted by $A\cup{B}$, is the set of all elements that are contained in either $A$ or $B$. For example, consider the same sets \eqref{eq:B4V9tinT2Y5LHf7E2Q7} as above. The union of $A$ and $B$ can be visualized in a Venn diagram like so:

Here, the shaded area in light blue represents the union of the two sets.

Disjoint sets

Recall that sets $A$ and $B$ are called disjoint sets if they have no common elements. For example, consider the following two disjoint sets:

$$\begin{aligned} A&=\{1,2,3\}\\ B&=\{8,9\}\\ \end{aligned}$$

Disjoint sets are represented in a Venn diagram like so:

Notice how unlike the previous Venn diagrams, the circles do not overlap, which means that the sets do not share any common element.

Universal and complementary sets

Recall that the universal setlink contains all elements under consideration. For instance, suppose we roll a dice - the universal set $S$ could hold all possible outcomes:

$$S=\{1,2,3,4,5,6\}$$

Now, suppose we have a set $A$ that holds only the even outcomes:

$$A=\{2,4,6\}$$

The Venn diagram in this case would look like the following:

The universal set $S$, as the name suggests, contains all the elements in the Venn diagram. Clearly, set $A$ is a subset of $S$.

A complementary set of $A$, which is denoted by $A'$ (or $A^c$ in some textbooks), holds all elements in the universal set $S$ that is not contained in $A$. This means that a complementary set contains everything else. In our dice example, the complementary set of $A$ will hold odd outcomes:

$$A'=\{1,3,5\}$$

In our Venn diagram above, the complementary set $A'$ is the set outside of $A$. To make this clearer, the light-blue region below represents $A'$:

Note that by definition, $A'$ will again be a subset of the universal set $S$.

Published by Isshin Inada

Edited by 0 others

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