A $\textbf{set}$ is a well deﬁned collection of objects considered as a whole. These objects are called $\textbf{elements}$
or $\textbf{members}$ of a set, and they can be anything, including numbers, letters, people, cities, and even other
sets. By convention, sets are usually denoted by capital letters and can be described or deﬁned by listing its
elements and surrounding the list by curly braces. For example, we can describe the set $A$ to be the set whose
members are the ﬁrst ﬁve prime numbers, or we can explicitly write: $A = \{2, 3, 5, 7, 11\}$. If $x$ is an element
of $A$, we write $x\in A$. Similarly, if $y$ is not an element of $A$, then we write $y\notin A$. Two sets $A$ and $B$ are said
to be equal, written as $A = B$, if they have the same elements. The order and repetition of elements do not
matter, so {red, white, blue} = {blue, white, red} = {red, white, white, blue}. Sometimes, more complicated
sets can be deﬁned by using a different notation. For example, the set of all rational numbers denoted by $\mathbb{Q}$
can be written as: $\{\frac{a}{b} \, \vert \, a, \, b \text{ are integers}, \, b \neq 0\}$. In English, this is read as “the set of all fractions such that
the numerator is an integer and the denominator is a non-zero integer."

Intro

We can also talk about the size of a set, or its cardinality. If $A = {1,2,3,4}$, then the cardinality of $A$, denoted
by $|A|$, is $4$. It is possible for the cardinality of a set to be $0$. This set is called the empty set, denoted by the
symbol $\emptyset$. A set can also have an inﬁnite number of elements, such as the set of all integers, prime numbers,
or odd numbers.

Cardinality

If every element of a set $A$ is also in a set $B$, then we say that $A$ is a subset of $B$, written $A\subseteq B$, or $A$
is contained in $B$. We can also write $B\supseteq A$, meaning that $B$ is a superset of $A$, or $B$ contains $A$. A proper
subset is a set $A$ that is strictly contained in $B$, written as $A\subset B$, meaning that $A$ excludes at least one element
of $B$. For example, consider the set $B = \{1,2,3,4,5\}$. Then $\{1,2,3\}$ is both a subset and a proper subset
of $B$, while $\{1,2,3,4,5\}$ is a subset but not a proper subset of $B$. Here are a few basic properties regarding
subsets:


<ul>
<li>The empty set is a proper subset of any nonempty set $A$: $\emptyset \subset A$</li>
<li>The empty set is a subset of every set $B$: $\emptyset \subseteq B$</li>
<li>Every set $A$ is a subset of itself: $A \subseteq A$</li>
</ul>

Subsets and Proper Subsets

The intersection of a set $A$ with a set $B$, written as $A \cap B$, is a set of all elements which are members of
both $A$ and $B$. Two sets are said to be disjoint if $A \cap B = \emptyset $. The union of a set $A$ with a set $B$, written as
$A \cup B$, is a set of all elements which are either members of $A$ or $B$. For example, if $A$ is the set of all positive
even numbers, and $B$ is the set of all positive odd numbers, then $A \cap B = \emptyset$, and $A \cup B = \Z^{+}$, or the set of all positive integers. 
Here are a few properties of intersections and unions:


<ul>
<li>$A \cup B = B \cup A$</li>
<li>$A \cup \emptyset = A$</li>
<li>$A \cap B = B \cap A$</li>
<li>$A \cap \emptyset = \emptyset$</li>
</ul>

Intersections and Unions

If $A$ and $B$ are two sets, then the relative complement of $A$ in $B$, written as $B - A$ or $B \setminus A$, is the set of
elements in $B$, but not in $A$: $B \setminus A = \{x \in {B} \,| \ x \notin A \}$. For example, if $B = \{1,2,3\}$ and $A = \{3,4,5\}$, then
$B \setminus A = \{1,2\}$. For another example, if $\R$ is the set of real numbers and $\Q$ is the set of rational numbers, then
$\R \setminus \Q$ is the set of irrational numbers. Here are some important properties of complements:


<ul>
<li>$A \setminus A = \emptyset$</li>
<li>$A \setminus \emptyset = A$</li>
<li>$\emptyset \setminus A = \emptyset$</li>
</ul>

Complements

In mathematics, some sets are referred to so commonly that they are denoted by special symbols. Some of
these numerical sets include:


<ul>
<li>$\mathbb{P}$ denotes the set of all prime numbers: $\{2,3,5,7,11,...\}$.</li>
<li>$\N$ denotes the set of all natural numbers: $\{0,1,2,3,...\}$.</li>
<li>$\Z$ denotes the set of all integer numbers: $\{...,-2,-1,0,1,2,...\}$.</li>
<li>$\Q$ denotes the set of all rational numbers: $\{\frac{a}{b} \,| \, a,b \in \Z, \, b \neq 0\}$</li>
<li>$\R$ denotes the set of all real numbers.</li>
<li>$\I$ denotes the set of all complex numbers.</li>
</ul>

In addition, the Cartesian product (also called the cross product) of two sets $A$ and $B$, written as $A \times B$,
is the set of all pairs whose ﬁrst component is an element of $A$ and whose second component is an element
of $B$. In set notation, $A \times B = \{(a;b) \, | \ a \in A,b \in B\}$. For example, if $A = \{1,2,3\}$ and $B = \{u,v\}$, then
$A \times B$ = $\{(1,u), \, (1,v), \, (2,u), \, (2,v), \, (3,u), \,(3,v)\}$. Given a set $S$, another signiﬁcant set is the power set of $S$,
denoted by $\mathcal P \left({S}\right)$, is the set of all subsets of $S: \{T \, | \ T \subseteq S \}$. For example, if $S = \{1,2,3\}$, 
then the power set of $S$ is: $\mathcal P \left({S}\right) = \{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$. 
It is interesting to note that, if $|S| = k$, then $|\mathcal P \left({S}\right)| = 2^k$

1 Lec23: Introduction to Sets