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% Document Info
\author{Aafiya}
\pdfinfo{
  /Title (grade-12-mathematics.pdf)
  /Creator (Cheatography)
  /Author (Aafiya)
  /Subject (Grade 12 Mathematics Cheat Sheet)
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\noindent
\begin{multicols}{3}
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    \vspace{-7pt}
    {\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{Grade 12 Mathematics Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Aafiya} via \textcolor{DarkBackground}{\uline{cheatography.com/171724/cs/36053/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Aafiya \\
  \uline{cheatography.com/aafiya} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 12th December, 2022.\\
   Page {\thepage} of \pageref{LastPage}.
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Relations}}  \tn
% Row 0
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\mymulticolumn{1}{x{5.377cm}}{DEFINITION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A relation R, from a non-empty set A to another non-empty set B is mathematically as an subset of A × B.   Equivalently, any subset of A × B is a relation from A to B. Thus,  R is a relation from A to B Û R Í A × B  Û R Í \{(a, b) : a Î A, b Î B\}} \tn 
% Row Count 7 (+ 7)
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\mymulticolumn{1}{x{5.377cm}}{DOMAIN OF A RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Let R be a relation from A to B. The domain of relation R is the set of all those elements   a Î A such that (a, b) Î R " b Î B. Thus,  Dom.(R) = \{a Î A : (a, b) Î R " b Î B\}. That is, the domain of R is the set of first components of all the ordered pairs which belong to R.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{RANGE OF A RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Let R be a relation from A to B. The range of relation R is the set of all those elements   b Î B such that (a, b) Î R " a Î A. Thus,  Range of R = \{b Î B : (a, b) Î R " a Î A\}. That is, the range of R is the set of second components of all the ordered pairs which belong to R.} \tn 
% Row Count 22 (+ 8)
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\mymulticolumn{1}{x{5.377cm}}{CO-DOMAIN OF A RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Let R be a relation from A to B. Then B is called the co-domain of the relation R. So we can observe that co-domain of a relation R from A into B is the set B as a whole.} \tn 
% Row Count 27 (+ 5)
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\mymulticolumn{1}{x{5.377cm}}{REFLEXIVE RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A relation R defined on a set A is said to be reflexive if a R a " a Î A i.e., (a, a) Î R  " a Î A} \tn 
% Row Count 31 (+ 4)
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\par\addvspace{1.3em}

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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Relations (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{SYMMETRIC RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A relation R defined on a set A is symmetric if (a, b) Î R Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa (i.e., whenever aRb then bRa).} \tn 
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\mymulticolumn{1}{x{5.377cm}}{TRANSITIVE RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A relation R on a set A is transitive if (a, b) Î R and (b, c) Î R Þ (a, c) Î R i.e., aRb and bRc Þ aRc.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{EQUIVALENCE RELATION} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Let A be a non-empty set, then a relation R on A is said to be an equivalence relation if (i) R is reflexive i.e., (a, a) Î R " a Î A i.e., aRa. (ii) For Let R is symmetric i.e., (a, b) Î R Þ (b, a) Î R " a, b Î A i.e., aRb Þ bRa. (iii)R is transitive i.e., (a, b) Î R and  (b, c) Î R Þ (a, c) Î R " a, b, c Î A i.e., aRb and bRc Þ aRc} \tn 
% Row Count 17 (+ 9)
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Functions}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{One-one function (Injective function or Injection)} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A function f : A ® B is one-one function or injective function if distinct elements of A have distinct images in B.   Thus,  f : A ® B is one-one Û f(a) = f(b) Þ a = b, "a, b Î A Û a ≠ b Þ f(a) ≠ f(b) " a, b Î A.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Onto function (Surjective function or Surjection)} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A function f : A ® B is onto function or a surjective function if every element of B is the f - image of some element of A. That implies f(A) = B or range of f is the co-domain of f. Thus, f : A ® B is onto Û f(A) = B i.e., range of f = co-domain of f.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{One-one onto function (Bijective function or Bijection)} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A function f : A ® B is said to be one-one onto or bijective if it is both one-one and onto i.e., if the distinct elements of A have distinct images in B and each element of B is the image of some element of A.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Matrices}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{PROPERTIES OF TRANSPOSE OF MATRICES :}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{(i)(A + B)T = AT + BT} \tn 
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\mymulticolumn{1}{x{5.377cm}}{(ii)(AT )T = A} \tn 
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\mymulticolumn{1}{x{5.377cm}}{(iii)(kA)T = kAT, where k is any constant} \tn 
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\mymulticolumn{1}{x{5.377cm}}{(iv) (AB)T = BT AT (v) (ABC)T = CT BT AT} \tn 
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% That's all folks
\end{multicols*}

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