Title: Theory / Unimportant / CS / Algo / Graphics / Fundamentals -Shirley Marschner{ / 02 Math
Tags: LaTeXError hasSeenConjDef
On every note of this deck, unless a note says otherwise:
Chapter: 2

Name: A mapping
Definition: something which takes an argument of one type and return an object of a particular type
Page: 13
Section: 1
Name2: A function
Denoted by: \(f\)

Name: \(a\) is a member of \(S\)
Page: 13
Section: 1
Notation: \(a\in S\)

Name: The cartesian products of \(A\) and \(B\)
Definition: The set of all ordered pairs \((a,b)\) with \(a\in A\) and \(b\in B\)
Page: 13
Section: 1
Variables: \(A,B\) sets
Notation: \(A\times B\)

Definition: \(A\times A\)
Page: 13
Section: 1
Variables: \(A\) set
Notation: \(A^2\)

Definition: The set of real numbers
Page: 14
Section: 1
Notation: \(\mathbb R\)

Definition: The set of non-negative real numbers
Page: 14
Section: 1
Notation: \(\mathbb R^+\)

Definition: The set of integers
Page: 14
Section: 1
Notation: \(\mathbb Z\)

Definition: The set of 3D points on the unit sphere
Page: 14
Section: 1
Notation: \(S^2\)

Definition: with domain \(A\)
Page: 14
Section: 1
Notation: \(f:A\mapsto B\)
Definition2: and target \(B\)
Conjdef: A function \(f\)

Name: The image of \(a\) by \(f\)
Definition: \(f(a)\)
Page: 14
Section: 1
Variables: \(f:A\mapsto B\),
\(a\in A\)

Name: The image of \(A'\)
Definition: the set of image of \(a\in A'\)
Page: 14
Section: 1
Variables: \(f:A\mapsto B\),
\(A'\subseteq A\)
Notation: \(f(A')\)

Name: The range of \(A\)
Definition: The image of \(A\)
Page: 14
Section: 1
Variables: \(f:A\mapsto B\),
Typ: Subset of \(B\)

Name: An injection
Definition: A function \(f:A\mapsto B\) such that each \(b\in B\) has at most one \(a\) such that \(f(a)=b\)
Page: 14
Section: 1
Variables: \(A,B\) sets

Name: A surjection
Definition: A function \(f:A\mapsto B\) such that each \(b\in B\) has at least one \(a\) such that \(f(a)=b\)
Page: 14
Section: 1
Variables: \(A,B\) sets

Name: A bijection
Definition: A function which is an injection and a surjection
Page: 14
Section: 1

Name: The inverse of \(f\)
Definition: [$]\defFun{B}A{b}{\text{the only $a$ such that $f(a)=b$}}[/$]
Page: 14
Section: 1
Variables: \(A,B\) sets
\(f:A\mapsto B\)
Notation: \(f^{-1}\)
Assuming: \(f\) is a bijection

Definition: \(a\le r\le b\)
Page: 15
Section: 1.2
Variables: \(a<b\) reals,
\(\langle \) being \((\) or \([\)
\(\rangle \) being \()\) or \(]\)
Notation: \(\langle a,b\rangle\)
Definition2: \(a<r\) if \(\langle\) is \((\)
Conjdef: The set of reals \(r\) such that
Definition3: \(r<b\) if \(\rangle\) is \()\)

Name: The intersection of \(A\) and \(B\)
Definition: The set of elements belonging to \(A\) and \(B\)
Page: 15
Section: 1.2
Variables: \(A,B\) sets
Notation: \(A\cap B\)

Name: The union of \(A\) and \(B\)
Definition: The set belonging to \(A\) or to \(B\)
Page: 15
Section: 1.2
Variables: \(A,B\) sets
Notation: \(A\cup B\)

Name: The difference of \(A\) and \(B\)
Definition: The set of elements belonging to \(A\) and not to \(B\)
Page: 16
Section: 1.2
Variables: \(A,B\) sets
Notation: \(A- B\)

Name: The logarithm in base \(a\) of \(x\)
Definition: the real \(y\) such that \(a^y=x\)
Page: 16
Section: 1.3
Variables: \(a>0\),
\(x\in\mathbb R\)
Notation: \(\log_ax\)
Definition2: \(\frac{\log x}{\log a}\)

Question: \(\log_a(xy)\)
Answer: \(\log_ax+\log_ay\)
Page: 16
Chapter: 2
Section: 1.3
Variables: \(a>0\),
\(x,y\in\mathbb R\)

Question: \(\log_a(x/y)=\)
Answer: \(\log_ax-\log_ay\)
Page: 16
Chapter: 2
Section: 1.3
Variables: \(a>0\),
\(x,y\in\mathbb R\)

Question: \(\log_a(x)\) expressed in base \(b\)
Answer: \(\log_ab\log_bx\)
Page: 16
Chapter: 2
Section: 1.3
Variables: \(a,b>0\),
\(x\in\mathbb R\)

Name: The natural logarithm
Definition: \(\log_e\)
Page: 16
Section: 1.3
Notation: \(\ln\)
Extra: neperien?

Question: [$]\dv{\log_ax}{x}[/$]
Answer: \(\frac{1}{x\ln a}\)
Page: 16
Chapter: 2
Section: 1.3

Question: [$]\dv{a^x}{x}[/$]
Answer: \(a^x\ln a\)
Page: 16
Chapter: 2
Section: 1.3

Name: A quadratic equation
Definition: An equation of the form \(Ax^2+Bx+C=0\)
Page: 17

Question: The number of real solution of \(P\) is
Answer: \(0\) if the discriminant is negative
\(1\) if it is 0
\(2\) if it is positive
Page: 17
Chapter: 2
Variables: \(A,B,C\in\mathbb R\)\(P(x)=Ax^2+Bx+C=0\) a quadratic equation

Question: The solution(s) are
Answer: \(\frac{-B\pm\sqrt{B^2-4AC}}{2A}\)
Page: 17
Chapter: 2
Variables: \(Ax^2+Bx+C=0\) a quadratic equation

Name: The discriminant of the equation \(Ax^2+Bx+C=0\)
Definition: \(B^2-4AC\)
Page: 17
Variables: \(Ax^2+Bx+C=0\) a quadratic equation

Name: A root of \(f\)
Definition: A value \(x\) such that \(f(x)=0\)
Page: 17
Variables: \(f(x)\) a function with target reals