Introduction

Table of contents

• Notation
• Sets, Functions, Images and pre-Images

Notation

The LaTeX command for each symbol will be given after their representation

• $\varnothing$ (\emptyset) denotes the empty set
• $\in$ (\in) denotes “is an element of”
• $\cup$ (\cup) and $\cap$ (\cap) denote union and intersection, respectively.
  • We will also use $\bigcup$ (\bigcup) and $\bigcap$ (\bigcap) for unions and intersections over families of sets.
• $\subset$ (\subset) denotes “is a subset of”
• $\mathbb{C}$ (\mathbb{C})) denotes the set of complex numbers, $\mathbb{R}$ (\mathbb{R}) the set of real numbers, $\mathbb{Q}$ (\mathbb{Q}) the set of rational numbers, $\mathbb{Z}$ (\mathbb{Z}) the set of integers, and $\mathbb{N}$ (\mathbb{N}) the set of natural numbers $\{1,2,3\}$. We also write ${\mathbb{R}}^{+}=\{x\in \mathbb{R}:x\ge 0\}$
• For sets $a$ and $B,$A \times B$ (\times) will denote their Cartesian product.
• For $n\ge 2$, $(R{)}^n$ is the Cartesian product of $\mathbb{R}$ with itself $n$ times
• Let $A_{\alpha },\alpha \in Y$ be a collection of sets indexed by an arbitrary set $Y$. Here, the product ${\prod }_{\alpha \in Y}{A}_{\alpha }$ (\prod_{\alpha \in Y} A_\alpha) is interpreted as the set of all functions $f:\mathcal{A}\to {\bigcup }_{\alpha \in Y}{A}_{\alpha }$ (f: \mathcal{A} \to \bigcup_{\alpha \in Y} A_\alpha) with the property that $f(\alpha )\in A_{\alpha }$
• if $A_{\alpha }=X$ for all $\alpha \in Y$, then we can write the above product as $X^Y$ and note that it is the set of all functions $f:Y\to X$
• A set $A$ is countable if it is in bijection with a subset of $\mathbb{N}$ or, equivalently, if there is an injection from $A$ to $\mathbb{N}$. A countable set may be finite or infinite, in which case we call it countable infinite. $\varnothing$ is countable.

Sets, Functions, Images and pre-Images

definition Image

Let $X$ and $Y$ be sets and let $f:X\to Y$ be a function. The image $f(A)$ of a set $A\subset X$ is defined to be
$f(A)=\{f(x):x\in A\}$