Section 2: Functions

• A rule of assignment is a subset of the cartesian product $C\times D$ such that if $(c,d)\in r$ and $(c,d')\in r$ then $d=d'$ .
  • The domain of $r$ is the set $\{x\in C|(x,y)\in r\mbox{ for some }y\}$ .
  • The image set of $r$ is the set $\{y\in D|(x,y)\in r\mbox{ for some }x\}$ .
• A function is a rule of assignment $r$ together with a set $B$ that contains the image set of $r$ as a subset.
  • The range of $f$ is the set $B$ .
• $f:A\rightarrow B$ is used to indicate that $A$ is the domain of $f$ and $B$ is the range of $f$ .

1. $f$ preserves inclusions and unions: $A'\subset A"\Rightarrow f(A')\subset f(A")$ , $f(\cup)=\cup f$ , $f(\cap)\subset\cap f$ , $f(A'-A")\supset f(A')-f(A")$ .
   1. The last two are equalities if $f$ is injective, i.e. in this case it preserves all four operations.
2. $f^{-1}$ preserves inclusions, unions, intersections and differences: $B'\subset B"\Rightarrow f^{-1}(B')\subset f^{-1}(B")$ , $f^{-1}(\cup)=\cup f^{-1}$ , $f^{-1}(\cap)=\cap f^{-1}$ , $f^{-1}(A'-A")=f^{-1}(A')-^{-1}f(A")$ .
3. $A'\subset f^{-1}(f(A'))$ ($=$ if $f$ is injective), $f(f^{-1}(B'))\subset B'$ ($=$ if $f$ is surjective).
4. If $g\circ f$ is injective then $f$ in injective and $g$ is injective on $f(A)$ .
   1. If $f$ has a left inverse then $f$ is injective.
5. If $f\circ h$ is surjective then $f$ is surjective, moreover, it is surjective on $h(D)\subset A$ .
   1. If $f$ has a right inverse then $f$ is surjective.
6. In general, a function $f$ may have more than one left inverse or more than one right inverse, but if it has both a left inverse and a right inverse then $f$ is bijective and both inverse functions are equal, therefore, the inverse is unique.