functions

title

description

• a function $f:S\to T$ is a mapping from set S to elements in set T, to which each element in 1 set (domain set) is mapped to exactly 1 element in another set (codomain set)
• $f:S\to T$
  • f: function
  • S: domain
  • T: codomain
  • f is a relation between S and T. For each $s$ in S there is a unique element $t$ in T so that $(s,t)\in R$
  • image / range of $f$ is set of all values $f$ can produce
• note:
  • image: set of output that the function actually produce for a particular input
  • range: set of output that the function could produced, eh

types of function

• injection
• surjection
• bijection

Def 7 (Function Composition)

• for $f:A\to B$ and $g:B\to C$
• $g\circ f=g(f(x))$
• $f\circ g=f(g(x))$
• $g\circ f\ne f\circ g$, if well defined

identity function

• identity function is a function returning the same output as its input
• $i{d}_A:A\to A$ - a function maps set A to itself
• eg
  • A is set of real numbers
  • $x\in A$, $id(x)=x$; for eg $x=5$, then $id(5)=5$
• prove
  • for every x in domain f, $f(x)=x$

left / right inverse

• $f\circ g=id$ ⇒ f is left inverse of g, g is right inverse of f

two-sided inverse

• f is both left and right inverse of g → f and g are two sided inverses of each other

total function

• f from A to B is total when f(x) exists for all $x\in A$

Claim 13, 14, 15: work for both sides

• f is injective and total ⇐> f has left inverse
• f is surjective ⇐> f has right inverse
• f is bijective ⇐> f has a two-sided inverse

Cantor-Bernstein-Schroeder

• if $f:S\to T$ is injective (or $|S|\le |T|$) and $g:T\to S$ (or $|T|\le |S|$) is injective ⇒ there is a bijective $h:S\to T$