Set and Function

Set

The collection of individuals.

Tpical kinds of sets

\(\mathbb{N}\): natural number

\(\mathbb{O}\): rational number

\(\mathbb{I}\): irrational number

\(\mathbb{R}\): real number

\[\mathbb{R}^{2} = \{(x, y) \vert x, y: \mathbb{R} \}\] \[\mathbb{R}^{3} = \{(x, y, z) \vert x, y, z: \mathbb{R} \}\] \[(a, b) = \{x \in \mathbb{R} \vert a<x<b \}\] \[[a ,b] = \{x \in \mathbb{R} \vert a \leq x \leq b \}\] \[[a, \infty) = \{x \in \mathbb{R} \vert a \leq x \}\]

\(\varnothing\): empty set

ex) \(A = \{x \in A \vert x \notin A \}\)

\(A^{c} = \{x \vert x \notin A \}\): Complement of A

Neighborhood, Boundary Point

A: set neighborhood of \(x_{0}\) in \(\epsilon\): points neighboring \(x_{0}\) For a positive number \(\epsilon\), \(\epsilon\) neighborhood of \(x_{0]\) is the set of all points the distance from \(x_{0}\) is within \(\epsilon\).

Here, \(\epsilon\) should be positive (>0) since neighborhood of some point could not be only the point itself.

\[= \{x \vert \lvert x - x_{0} \rvert < \epsilon \}\]

y: boundary point of A. Any neighborhoood of point y contains at least one point in A and at least one point in complement of A (meets A and its comlement at the same time) Regardless of \(\epsilon\) size, if y is a boundary point of A, every \(\epsilon\) neighborhood of y meets both of \(A\) and \(A^{c}\).

\[\{All \ \epsilon\ neighborhood\ of\ y \}\cap A \neq \varnothing \ \{All \ \epsilon\ neighborhood\ of\ y \}\cap A^{c} \neq \varnothing\]

A=(a, b) a, b: boundary points of A

A: open set (a, b): open interval. Kind of open set a, b: not contained in A Open set is

\[x \in A \ \{Some \ \epsilon\ neighborhood\ of \ x \} \subset A\]

A: closed set

a, b: only boundary points of A, contained in the set A.

Closed set is the set containing all of its boundary points.

\[\{All \ boundary \ points \ of \ A\} \subset A\]

Bounded Set

Consists of points whose distance from the origin is finite.

The set is bounded set if its every point from the origin has finite distance in xy plane.

However, x-axis and y-axis are not the bounded sets since they are extended to infinity.

Function

Function defined in a set.

A, B: set

f: function to match each x to unique point f(x) in B

x: \(x \in A\)

A is “domain of f”, B is “co-domain of f”

x is “independent variable”, f(x) is “dependent variable”

When \(A, B \subset \mathbb{R}\), y =f(x)

“Range of f”: \(\{f(x) \vert x \in A \} \subset B\)

Types of Functions

1. Constant function

\[f(x)=c, \ x\in A\]

The function matching every point in the domain to a fixed number (constant).

1. Identity function

\[f= id_{A}\] \[f: A \rightarrow A\] \[f(x)=x, \ x\in A\]

The function matching every point to itself.

1. One-to-one function

\[x_{1} \neq x_{2}, \ x_{1} \in A, \ x_{2} \in A \ \Rightarrow f(x_{1}) \neq f(x_{2])\]

The function that any two different points of domain correspond to the different values.

1. Onto function \(f: A \rightarrow B\)

B = “Range of f”, (a.k.a “co-domain of f” = “range of f”)

Composition of Functions

\[A \overset{f}{\rightarrow} B \overset{g}{\rightarrow} C\] \[x \overset{f}{\rightarrow} f(x) \overset{g}{\rightarrow} g(f(x))\] \[x \overset{g\cdot f}{\rightarrow} g(f(x))\]

\(g\cdot f\): composition of g and f

\[(g\cdot f)(x)= g(f(x))\]

In general, \(g\cdot f \neq f\cdot g\)

Inverse Function

\[A \overset{f}{\rightarrow} B \overset{f^{-1}}{\rightarrow} A\] \[x \rightarrow f(x) \rightarrow x\] \[x \overset{f^{-1}}{\leftarrow} f(x)\]

Necessary condition for f inverse funtion \(f^{-1}\)

f: one-to-one and onto function

\[x \overset{f^{-1}\cdot f}{\rightarrow} x\] \[f^{-1}\cdot f(x)= x \ \Rightarrow f^{-1}\cdot f=id_{A}\]

y=f(x)

\[f\cdot f^{-1}(y)=y \ \Rightarrow f\cdot f^{-1}= id_{B}\] \[y=f(x) \rightarrow x=f^{-1}(y)\]

We exchange x and y such as, \(y=2x+1 \rightarrow x=\frac{y-1}{2} \rightarrow y=\frac{x-1}{2}\) Graph y=2x+1 and y=(x-1)/2 are symmetric about y=x

Elementary functions

1. \(f(x)=c\): Constant function
2. \(f(x)=x^{n}\): Monomial function
3. \(f(x)=a_{0}+ a_{1}x+ ...+a_{n}x^{n}, \ a_{n}\neq 0\): Polynomial function P(x)
4. \(\frac{P_2(x)}{P_1(x)}, \ P_1(x)\neq 0\): Rational function
5. \(\sqrt{P(x)}\): Irrational function
6. \(y=a^x, \ a>0\): Exponential function
7. \(y=log_{a}x\): Logarithmic function. Inverse function of exponential function.
8. \(\sin x, \cos x, \tan x, \csc x, \sec x, \cot x\): Trigonometric functions \(\csc x = \frac{1}{\sin x}, \sec x= \frac{1}{\cos x}, \cot x=\frac{1}{\tan x}\)
9. \(\sin^{-1}x, \cos^{-1}x, \tan^{-1}x\): Inverse Trigonometic function 10.\(\sinh x=\frac{e^{x}-e^{-x}}{2}, \cosh x=\frac{e^{x}+e^{-x}}{2}, \tanh x =\frac{\sinh x}{\cosh x}\) : Hyperbolic function

\(\sin x, \cos x, \tan x, \csc x, \sec x, \cot x\): Trigonometric functions

\[\csc x = \frac{1}{\sin x}, \sec x= \frac{1}{\cos x}, \cot x=\frac{1}{\tan x}\]

source “KMOOC 집콕 미적학학 1: 활용을 중심으로”

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