\displaystyle \textbf{1.}~\text{Let }A\text{ and }B\text{ be two non-empty sets. Then, a subset }f\text{ of }A\times B\text{ is a function from }A\text{ to }B,\text{ if}
\displaystyle \text{(i) for each }a\in A\text{ there exists }b\in B\text{ such that }(a,b)\in f
\displaystyle \text{(ii) }(a,b)\in f\text{ and }(a,c)\in f\Rightarrow b=c.
\displaystyle \text{In other words, a subset }f\text{ of }A\times B\text{ is a function from }A\text{ to }B,\text{ if each element of }A
\displaystyle \text{appears in some ordered pair in }f\text{ and no two ordered pairs in }f\text{ have the same first element.}

\displaystyle \textbf{2.}~\text{Let }A\text{ and }B\text{ be two non-empty sets. Then, a function }f\text{ from }A\text{ to }B\text{ associates every element}
\displaystyle \text{of }A\text{ to a unique element of }B.\text{ The set }A\text{ is called the domain of }f\text{ and the set }B\text{ is known as its}
\displaystyle \text{co-domain. The set of images of elements of set }A\text{ is known as the range of }f.

\displaystyle \textbf{3.}~\text{If }f:A\to B\text{ is a function, then }x=y\Rightarrow f(x)=f(y)\text{ for all }x,y\in A.

\displaystyle \textbf{4.}~\text{A function }f:A\to B\text{ is a one-one function or an injection, if}
\displaystyle f(x)=f(y)\Rightarrow x=y\text{ for all }x,y\in A\text{ or, }x\neq y\Rightarrow f(x)\neq f(y)\text{ for all }x,y\in A.
\displaystyle \text{Graphically, if the graph of a function does not take a turn, in other words a straight line}
\displaystyle \text{parallel to }x\text{-axis does not cut the curve at more than one point, then it is a one-one function.}
\displaystyle \text{Note that a function is one-one, if it is either strictly increasing or strictly decreasing.}

\displaystyle \textbf{5.}~\text{A function }f:A\to B\text{ is an onto function or a surjection, if range}(f)=\text{co-domain}(f).

\displaystyle \textbf{6.}~\text{Let }A\text{ and }B\text{ be two finite sets and }f:A\to B\text{ be a function.}
\displaystyle \text{(i) If }f\text{ is an injection, then }n(A)\le n(B).
\displaystyle \text{(ii) If }f\text{ is a surjection, then }n(A)\ge n(B).
\displaystyle \text{(iii) If }f\text{ is a bijection, then }n(A)=n(B).

\displaystyle \textbf{7.}~\text{If }A\text{ and }B\text{ are two non-empty finite sets containing }m\text{ and }n\text{ elements respectively, then}
\displaystyle \text{(i) Number of functions from }A\text{ to }B=n^{m}.
\displaystyle \text{(ii) Number of one-one functions from }A\text{ to }B=\begin{cases}{}^{n}C_{m}\,m!,&\text{if }n\ge m\\0,&\text{if }n<m\end{cases}
\displaystyle \text{(iii) Number of onto functions from }A\text{ to }B=\begin{cases}\sum_{r=1}^{n}(-1)^{\,n-r}\,{}^{n}C_{r}\,r^{m},&\text{if }m\ge n\\0,&\text{if }m<n\end{cases}
\displaystyle \text{(iv) Number of one-one and onto functions from }A\text{ to }B=\begin{cases}n!,&\text{if }m=n\\0,&\text{if }m\neq n\end{cases}

\displaystyle \textbf{8.}~\text{If a function }f:A\to B\text{ is not an onto function, then }f:A\to f(A)\text{ is always an onto function.}

\displaystyle \textbf{9.}~\text{The composition of two bijections is a bijection.}

\displaystyle \textbf{10.}~\text{If }f:A\to B\text{ is a bijection, then }g:B\to A\text{ is inverse of }f,\text{ iff }f(x)=y\Rightarrow g(y)=x
\displaystyle \text{or, }g\circ f=I_{A}\text{ and }f\circ g=I_{B}.

\displaystyle \textbf{11.}~\text{Let }f:A\to B\text{ and }g:B\to A\text{ be two functions.}
\displaystyle \text{(i) If }g\circ f=I_{A}\text{ and }f\text{ is an injection, then }g\text{ is a surjection.}
\displaystyle \text{(ii) If }f\circ g=I_{B}\text{ and }f\text{ is a surjection, then }g\text{ is an injection.}

\displaystyle \textbf{12.}~\text{Let }f:A\to B\text{ and }g:B\to C\text{ be two functions. Then}
\displaystyle \text{(i) }g\circ f:A\to C\text{ is onto }\Rightarrow g:B\to C\text{ is onto.}
\displaystyle \text{(ii) }g\circ f:A\to C\text{ is one-one }\Rightarrow f:A\to B\text{ is one-one.}
\displaystyle \text{(iii) }g\circ f:A\to C\text{ is onto and }g:B\to C\text{ is one-one }\Rightarrow f:A\to B\text{ is onto.}
\displaystyle \text{(iv) }g\circ f:A\to C\text{ is one-one and }f:A\to B\text{ is onto }\Rightarrow g:B\to C\text{ is one-one.}

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