% This is lecture no. 1 of MH1300/: Foundations Of Mathematics.

\chapter{Speaking Mathemagically}

\section{Variables}

\begin{definition}
	Variable.
	A variable is a symbol in an expression that may be replaced by the name of any element of a given set, representing an unspecified member of a class of objects, numbers, etc.
\end{definition}

\subsection{Mathematical Statements}

\bigbreak
\noindent
Types of mathematical statements:

\begin{itemize}
	\item Universal Statements: Statement that says that a certain property is true for all elements in a set
		\begin{exmp}
			All integers are divisible by 1.
		\end{exmp}
	\item Conditional Statements: Statement that says that if one thing is true, then some other thing is also true.
		\begin{exmp}
			If $x$ is divisible by 1300, then  $x$ is divisible by 13.
		\end{exmp}
	\item Existential Statements: Statement that says that given a property that may or may not be true, there is at least one thing for which the property is true.
		\begin{exmp}
			There is a prime number between 1000 and 2000.
		\end{exmp}
	\item Universal Conditional Statement:	Statement that is both universal and conditional.
		\begin{exmp}
			For all objects $J$, if  $J$ is a square then  $J$ has four sides.
		\end{exmp}
	\item Universal Existential Statements: Statement that is universal because its first part says that a certain property is true for all objects of a given tye, and is existential because its second part asserts the existence of something.
		\begin{exmp}
			Every ring contains an additive identity, which can be rewritten as: For all rings $R$, this is an element $x$ such that $x$ is as additive identity contained in $R$.
		\end{exmp}
	\item Existential Universal Statements: Statement that is existential because its first pasrt asserts that a certain object exists, and is universal because its second part says that the object satisfies a certain property for all things of a certain kind.
		\begin{exmp}
			There is a student of MH1300 who sleeps during all lectures conducted in LT23. Which can be rewritten as: There is a student $x$ of MH1300 such that $x$ sleeps during all lectures conducted in LT23.
		\end{exmp}
\end{itemize}

\section{The Language of Sets}

\begin{definition}
	Set.
	A set is an unordered collection of objects.
	The objects inside a set are called elements or members of a set.
	Two sets are equal if and only if they have exactly the same elements.
\end{definition}

\subsection{Notation}

\bigbreak
\noindent
There are some basic notations for sets:

\begin{itemize}
	\item Use uppercase letters to denote sets.
	\item An element $x$ is inside set $A$ denoted by $x \in A$
	\item An element $x$ is not inside set $A$ denoted by $x \notin A$
	\item A way to define a set is to indicate its elements between a pair of braces, i.e. \underline{Set-roster notation}.
	\begin{exmp}
		\{1,2,3\} \quad \{1,2,3,\ldots,50\} \quad \{5,10,15,20,\ldots\}
	\end{exmp}
\end{itemize}

\bigbreak
\noindent
The following notations are for \textbf{Special Sets}:

\begin{itemize}
	\item $\mathbb{N}$ : Set of natural numbers \{0,1,2,3,\ldots\}
	\item $\mathbb{Z}$ : Set of integers \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}
	\item $\mathbb{Q}$ : Set of rational numbers, or quotients of integers
	\item $\mathbb{R}$ : Set of real numbers
	\item $\mathbb{C}$ : Set of complex numbers
\end{itemize}

\subsubsection{Set-Builder Notation}

Let $S$ denote a set and let $P(x)$ be a property that elements of $S$ may or may not satisfy. We may define a new set to be the set of all elements $x$ is $S$ such that $P(x)$ is true.

\noindent
This is denoted by:
\[
	\{x \in S | P(x)\}
.\] which is read ``The set of all $x$ in $S$ such that $P(x)$."

\bigbreak
\noindent
Another example:

\begin{exmp}
	$\mathbb{Z}^{+} = \{x \in \mathbb{Z} | x > 0\}$ which is read: ``The set of all integers $x$ such that $x > 0$."
\end{exmp}

\subsection{Subsets}

\begin{definition}
	Subset.
	If $A$ and $B$ are sets , then $A$ is called a \textbf{subset} of $B$, written  $A \subseteq B$ \underline{if and only if} every element of $A$ is also an element of $B$.
\end{definition}

\bigbreak
\noindent
Symbolically:

$A \subseteq B$ means that for all elements $x$, if $x \in A$ then $x \in B$.

$A \notin B$ means that there is an element $x$ such that $x \in A$ and $x \notin B$.

\begin{definition}
	Proper Subset.
	Let $A$ and $B$ be sets. $A$ is a \textbf{proper subset} of $B$, written $A \subset B$, if and only if every element of $A$ is in $B$ but there is at least one element of $B$ that is not in $A$.
\end{definition}

\subsection{Cartesian Products}

\begin{definition}
	Ordered Pairs.
	Given elements $a$ and $b$, the symbol $(a, b)$ denotes the \textbf{ordered pair} consisting of  $a$ and $b$ together with the specification that:
	\begin{itemize}
		\item $a$ is the first element of the pair
		\item $b$ is the second element of the pair
	\end{itemize}
	Two ordered pairs $(a, b)$ and $(c, d)$ are \textbf{equal} if and only if $a=c$ and $b=d$.

	\bigbreak
	\noindent
	Symbolically:
	\[
		(a,b) = (c,d) \quad \text{means that} \quad a=c \quad \text{and} \quad b=d
	.\] 
\end{definition}

\begin{exmp}
	Is (1,2) = (2,1)?
	No, since they have different orders.
\end{exmp}

\begin{definition}
	Cartesian Products.
	Given two sets $A$ and $B$, the Cartesian product of $A$ and $B$, denoted $A\times B$ (read `$A$ cross $B$') is the set of all ordered pairs $\left(a,b\right)$, where $a$ is in $A$ and $b$ is in $B$.

	Symbolically:
	\[
		A \times B = \left\{\left(a,b\right)| a \in A \text{ and } b \in B\right\}
	.\] 
\end{definition}
