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Thread: Precalculus

  1. #1
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Feb 2008
    Yuma, AZ, USA


    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 $\displaystyle \emptyset$

    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".
    $\displaystyle \in $ is short for "element of" or "in".
    $\displaystyle \not \in $ is short for "not element of" or "not in".
    $\displaystyle \forall$ is short "for all" or "for every"
    $\displaystyle \subset$ is short for subset
    $\displaystyle \cup $ is short for "union" or "OR"
    $\displaystyle \cap $ is short for "intersection" or "AND"


    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 $\displaystyle P=\{\mbox{cat, rat} \}$. 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.
    $\displaystyle D=\{ 0,1,2,3,4,5,6,7,8,9\}$. Another way to write a set is called set builder notation. $\displaystyle D=\{ x| x \mbox{ is a digit }\}$. Both of these represent the same set. In general the It can be written as $\displaystyle S=\{x|f(x) \}$ 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
    $\displaystyle A=\{ 1,2,3\}$ is the same as $\displaystyle B=\{2,1,3 \}$. A and B represent the same set. We could write $\displaystyle 3 \in A$ would be read as "Three is an element of A" or "Three is in A". Two sets are equal if $\displaystyle A=B$ that is, if every element of A is in B and every element of B is in A. In symbols $\displaystyle \forall x \in A, x \in B$ and $\displaystyle \forall x \in B, x \in A$. There is another relationship that can exits between sets. $\displaystyle A \subset B$ A is a subset of B if for all x in A x is in B. In symbols $\displaystyle \forall x \in A x \in B$. There is a way we can combine sets called union. Let $\displaystyle C=\{ 3,5\}$ and $\displaystyle D=\{ 4,5,6\}$ $\displaystyle C \cup D =\{ 3,4,5,6\}$ The elements of C or D. Note: that the element five is only in the set once. $\displaystyle C \cap D =\{ 5\}$ Is the elements in C and D.

    Sets of Numbers:
    The Natrual number(sometimes called the counting numbers) $\displaystyle \mathbb{N}=\{ 1,2,3,...\}$ the ... called ellipsis tell us that the pattern continues indefinitely. The set of Integers is $\displaystyle \mathbb{Z}=\{ ...,-3,-2,-1,0,1,2,3,...\}$. The positive and negative Natural numbers and 0. The set of Rational Numbers $\displaystyle \mathbb{Q}=\left\{\frac{a}{b} \bigg|a,b \in \mathbb{Z} \mbox{ and } b \ne 0 \right\}$. Notice that $\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$. 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 $\displaystyle \pi,e,\sqrt{2}$. The set of Real Number $\displaystyle \mathbb{R}$ is the union of Rational and Irrational numbers. We will be using the set of real numbers. Any exceptions will be noted.

    In mathematics letters are often used to represent any member from a set of numbers (for us usually $\displaystyle \mathbb{R}$). 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 $\displaystyle \sqrt{5},\frac{\pi}{6}$, 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 $\displaystyle \mathbb{R}$. The number Zero cuts the number line into three parts. The positive Real Numbers $\displaystyle \mathbb{R}^+$ ,Zero, and the negative Real Numbers $\displaystyle \mathbb{R}^-$. 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.

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

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

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

    We can expand the above to greater than or equal to by saying if $\displaystyle a,b \in \mathbb{R}\\\ a \ge b $ if and only if $\displaystyle a-b \in \mathbb{R}^+\cup \{0\}$ Using a similar argument we can define less than or equal to.
    Last edited by TheEmptySet; May 28th 2008 at 06:19 PM.
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