Summary
Properties of a function
F1 -> every x has an arrow
F2 -> no x has more than one arrow
function -> every x has only one out going arrow
Identity function
Existence of the inverse
onto (surjective) -> every y has at least one incoming arrow
1-1 (injective) -> no y has more than one incoming arrow
bijective -> both 1-1 and onto, every y has only one incoming arrow
Boolean functions
Proving properties
Concept
Image and range
- typical notation for functions
Equality between functions (Theorem 2.7)
The composition of two functions is a function (Theorem 2.8)
- intersecting codomain and domain must be the same/subset
Composition with identity (Corollary 2.9)
Exisitence of an inverse function
the inverse is well defined iff the function is a bijection
Inverse of a bijectve function is bijective (Theorem 2.10)
Composition with inverse is identity (Corollary 2.11)
Composition preserves onto, 1-1 and bijectivity (Theorem 2.12)
Application
Identifying funtions
Definition of sequences as functions
Extra
Tikz template for arrow diagram between number lines
\usepackage{tikz}
\usetikzlibrary{calc}
\newcommand\numline[3]% start, start val, end val
{
\draw[->] (#1,#2-0.5) {} -- (#1,#3+0.5) {};
%\foreach \y in {#2,...,#3}
%\node at (#1,\y) {a};
%\draw[-, shift={(#1,\y)}] (3pt,0pt) -- (-3pt,0pt);
}
\newcommand\marker[4][left]% left/right, coord, label
{
%\node[#1] at (#2) {};
%\draw[shift=(0,1)] (3pt,0pt) {} -- (-3pt,0pt) {};
\node[label=#1:\small#3] (#4#3) at (#2) {};
\draw ($(3pt,0)+(#2)$) -- ($(-3pt,0)+(#2)$);
}
\begin{document}
\begin{tikzpicture}[short/.style={shorten <=2pt,shorten >=2pt}]
\node (X) at (0,2.5) {X};
\node (R) at (2,2.5) {$i_x$};
\node (Y) at (4,2.5) {X};
\numline{0}{-1}{1}
\numline{4}{-1}{1}
\marker{0,-1}{-1}
\marker{0,0}{0}
\marker{0,1}{1}
\marker[right]{4,-1}{-1}
\marker[right]{4,0}{0}
\marker[right]{4,1}{1}
\draw[->, short] (0,-1) -- (4,-1);
\draw[->, short] (0,0) -- (4,0);
\draw[->, short] (0,1) -- (4,1);
\end{tikzpicture}
\end{document}