Math Help - Precalculus

1. Precalculus

This is a work in progress. Please private message me with any comments or concerns. I will be adding sections as time permits.
Thanks TheEmptySet $\emptyset$

Preamble: These are common abbreviations "symbols" that will be used in this section. I will put new symbols with defintions at the beginning of each section.

| (vertical bar) is short for "such that".
$\in$ is short for "element of" or "in".
$\not \in$ is short for "not element of" or "not in".
$\forall$ is short "for all" or "for every"
$\subset$ is short for subset
$\cup$ is short for "union" or "OR"
$\cap$ is short for "intersection" or "AND"

Sets:

In mathematics we deal with collections of similar but distinct objects. A set is a collection of objects. For example, the set of my pets (P) would consist of $P=\{\mbox{cat, rat} \}$. This is called the roster method. (Each object is listed)The elements(obejcts) in the set are cat and rat and are enclosed by {}.

Lets look at another set. Let D be the set of all digits.
$D=\{ 0,1,2,3,4,5,6,7,8,9\}$. Another way to write a set is called set builder notation. $D=\{ x| x \mbox{ is a digit }\}$. Both of these represent the same set. In general the It can be written as $S=\{x|f(x) \}$ where f(x) is the condtion for membership of the set. In the above example f(x) is the statement that x is a digit.

Equality of Sets:

The order of the elements in a set does not matter. For example
$A=\{ 1,2,3\}$ is the same as $B=\{2,1,3 \}$. A and B represent the same set. We could write $3 \in A$ would be read as "Three is an element of A" or "Three is in A". Two sets are equal if $A=B$ that is, if every element of A is in B and every element of B is in A. In symbols $\forall x \in A, x \in B$ and $\forall x \in B, x \in A$. There is another relationship that can exits between sets. $A \subset B$ A is a subset of B if for all x in A x is in B. In symbols $\forall x \in A x \in B$. There is a way we can combine sets called union. Let $C=\{ 3,5\}$ and $D=\{ 4,5,6\}$ $C \cup D =\{ 3,4,5,6\}$ The elements of C or D. Note: that the element five is only in the set once. $C \cap D =\{ 5\}$ Is the elements in C and D.

Sets of Numbers:
The Natrual number(sometimes called the counting numbers) $\mathbb{N}=\{ 1,2,3,...\}$ the ... called ellipsis tell us that the pattern continues indefinitely. The set of Integers is $\mathbb{Z}=\{ ...,-3,-2,-1,0,1,2,3,...\}$. The positive and negative Natural numbers and 0. The set of Rational Numbers $\mathbb{Q}=\left\{\frac{a}{b} \bigg|a,b \in \mathbb{Z} \mbox{ and } b \ne 0 \right\}$. Notice that $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$. This tells us that all natural numbers of integers and that all integers are rational numbers.
We will assume the existance of a set called irrational numbers. Irrational numbers are numbers that cannot be written as fractions. (Non repeating non terminating deciamls) Examples are $\pi,e,\sqrt{2}$. The set of Real Number $\mathbb{R}$ is the union of Rational and Irrational numbers. We will be using the set of real numbers. Any exceptions will be noted.

[B]Variables:[\B]
In mathematics letters are often used to represent any member from a set of numbers (for us usually $\mathbb{R}$). We will use the convention that variables will come from the end of the alphabet. Ex: t,x,y,z. A constant is a fixed number, such as $\sqrt{5},\frac{\pi}{6}$, or a letter that represents a fixed, possibly unspecified, number. We will use the convention that constants will come from the beginning of the alphabet. Ex: a,b,c.

[B]Real Number Line:[\B]
A number line is a pictorial representation of the set of Real Numbers $\mathbb{R}$. The number Zero cuts the number line into three parts. The positive Real Numbers $\mathbb{R}^+$ ,Zero, and the negative Real Numbers $\mathbb{R}^-$. This is known as the Law of Trichotomy. Graphically on the number line positive numbers are to the right of zero, and negative numbers are to the left of zero.

$>$greater than symbol
$<$less than symbol
$\ge$greater or equal to symbol
$\le$less than or equal to symbol

Relations on Real numbers
Let $a,b \in \mathbb{R}$(a,b are both Real Numbers). We say b is greater than a if $b-a \in \mathbb{R}^+$ this can be written in $b > a$.
Let $c,d \in \mathbb{R}$ We say c is less than d if $c-d \in \mathbb{R}^-$ or $c < d$

Example 1: $17 > 6$ becuase $17-6=11 \in \mathbb{R}^+$

We can expand the above to greater than or equal to by saying if $a,b \in \mathbb{R}\\\ a \ge b$ if and only if $a-b \in \mathbb{R}^+\cup \{0\}$ Using a similar argument we can define less than or equal to.