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% Document Info
\author{mkenny}
\pdfinfo{
  /Title (discrete-math-proofs.pdf)
  /Creator (Cheatography)
  /Author (mkenny)
  /Subject (Discrete Math - Proofs Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Discrete Math - Proofs Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{mkenny} via \textcolor{DarkBackground}{\uline{cheatography.com/177372/cs/37028/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}mkenny \\
  \uline{cheatography.com/mkenny} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 15th February, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
  www.readability-score.com
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{DEFINITIONS}}  \tn
% Row 0
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Even Integer & An integer x is even if there is an integer k such that x = 2k. \tn 
% Row Count 3 (+ 3)
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Odd Integer & An integer x is odd if there is an integer k such that x = 2k+1. \tn 
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Parity & Whether the number is odd or even \tn 
% Row Count 8 (+ 2)
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Divides & An integer x divides an integer y if and only if x ≠ 0 and y = kx, for some integer k. Denoted x|y. If x does not divide y, then that fact is denoted x ∤ y.  If x divides y, then y is said to be a {\bf{multiple}} of x, and x is a {\bf{factor}} or {\bf{divisor}} of y. \tn 
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Prime & An integer n is prime if and only if n \textgreater{} 1, and the only positive integers that divide n are 1 and n. \tn 
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Rational & A number r is rational if there exist integers x and y such that y ≠ 0 and r = x/y. \tn 
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{\bf{ZERO}} & 0 is rational. For example if x = 0 and y = 1, then y ≠ 0 and x/y = 0/1 = 0. \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{METHOD DEFINITIONS}}  \tn
% Row 0
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\mymulticolumn{1}{x{5.377cm}}{constructive proof of existence} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A proof that shows that an existential statement is true.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{proof by exhaustion} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Allowed assumptions in proofs}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{The rules of algebra.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}For example if x, y, and z are real numbers and x = y, then x+z = y+z.} \tn 
% Row Count 3 (+ 3)
% Row 1
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\mymulticolumn{1}{x{5.377cm}}{The set of integers is closed under addition, multiplication, and subtraction.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}n other words, sums, products, and differences of integers are also integers.} \tn 
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% Row 2
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\mymulticolumn{1}{x{5.377cm}}{Every integer is either even or odd.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}This fact is proven elsewhere in the material.} \tn 
% Row Count 9 (+ 2)
% Row 3
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\mymulticolumn{1}{x{5.377cm}}{If x is an integer, there is no integer between x and x+1.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}In particular, there is no integer between 0 and 1.} \tn 
% Row Count 13 (+ 4)
% Row 4
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\mymulticolumn{1}{x{5.377cm}}{The relative order of any two real numbers.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}For example 1/2 \textless{} 1 or 4.2 ≥ 3.7.} \tn 
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% Row 5
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\mymulticolumn{1}{x{5.377cm}}{The square of any real number is greater than or equal to 0.} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}This fact is proven in a later exercise.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Choosing a Method}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mkenny_1676063732_undergrad_non-math-majors_resources_proof-method_0.JPG}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Common keywords and phrases in proofs}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{Thus, therefore then, hence, it follows that}} \newline % Row Count 1 (+ 1)
A statement that follows from the previous statement(s) \newline % Row Count 3 (+ 2)
{\emph{ex. n and m are integers. Therefore, n+m is also an integer.}} \newline % Row Count 5 (+ 2)
{\bf{Let, suppose}} \newline % Row Count 6 (+ 1)
Introduce a new variable \newline % Row Count 7 (+ 1)
{\emph{ex. "Let x be a positive integer" "Suppose that x is a positive integer"}} \newline % Row Count 9 (+ 2)
{\bf{Since}} \newline % Row Count 10 (+ 1)
If a statement depends on a fact that appeared earlier in the proof or in the assumptions of the theorem, it can be helpful to remind the reader of that fact before the statement.  \newline % Row Count 14 (+ 4)
{\emph{ex. "Since x \textgreater{} 0 and y \textgreater{} z, then xy \textgreater{} xz."}} \newline % Row Count 15 (+ 1)
{\bf{By definition}} \newline % Row Count 16 (+ 1)
A fact that is known because of a definition \newline % Row Count 17 (+ 1)
{\emph{ex. "The integer m is even. By definition, m = 2k for some integer k."}} \newline % Row Count 19 (+ 2)
{\bf{By assumption}} \newline % Row Count 20 (+ 1)
A fact that is known because of an assumption \newline % Row Count 21 (+ 1)
{\emph{ex. "By assumption, x is positive. Therefore x \textgreater{} 0."}} \newline % Row Count 23 (+ 2)
{\bf{"gives" and "yields" }} \newline % Row Count 24 (+ 1)
useful to say that one equation or inequality follows from another \newline % Row Count 26 (+ 2)
provides clarity to justify algebraic steps \newline % Row Count 27 (+ 1)
*ex. Multiplying both sides of the inequality x \textgreater{} y by 2 gives 2x \textgreater{} 2y. \newline % Row Count 29 (+ 2)
Substituting m = 2k into m2 yields (2k)2* \newline % Row Count 30 (+ 1)
} \tn 
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\vfill
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Common keywords and phrases in proofs (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{Since z \textgreater{} 0, we can multiply both sides of the inequality x \textgreater{} y by z to get xz \textgreater{} yz.% Row Count 2 (+ 2)
} \tn 
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% That's all folks
\end{multicols*}

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