Mapping

from University of Tokyo 1S1 Basic Mathematical Sciences: Linear Algebra Mapping

• Definition of Mapping
• Generalization of Functions
  • Seems interesting (blu3mo)
  • It’s like making the domain more general, right?
  • Honestly, there’s no difference (takker)
    • It’s just that people tend to call mappings where the domain and range are limited to numbers as functions, that’s about it
• For example, $f(x)=x^2$ is a mapping from $R$ to $R$
• Definition
  • Given sets X and Y,
  • A mapping $f:X\to Y$ from X to Y is defined as
    • A rule that assigns a unique element $f(x)$ in Y to each element $x$ in X
      • The uniqueness is crucial (blu3mo)
        • We learned this in IB as the definition of a function (blu3mo)
      • If it’s not unique, the notation $f(x)$ won’t make sense in the first place (takker)
• Examples
  • To represent $f(x)=x^2$ as a mapping,
    • It can be written as $$\begin{gathered} f:R\to R \\ x\mapsto x^2 \end{gathered}$$
    • Specify the entire set of inputs and outputs and then define them using \mapsto
  • For something less like traditional functions,
    • Addition is also a mapping
      • $$\begin{gathered} h:R^2\to R \\ (x,y)\mapsto x+y \end{gathered}$$
      • This represents a mapping from ℝ^2 (two dimensions) to ℝ (one dimension)
    • I see, it’s similar to how addition and other operations can be treated as functions in languages like Haskell using infix notation (blu3mo)
      • I thought so, but it’s actually different; the symbol for addition is not separately defined
        • It’s just wrapped
      • Well, addition is indeed a mapping from $R^2$ to $R$
    • I remember discussing these notations’ variations at the watercooler (takker)
      • I’ll add the link later
  • The Identity Mapping is a mapping where the input and output are all the same
    • In terms of definition,
    • $$\begin{gathered} i{d}_X:X\to X \\ x\mapsto x \end{gathered}$$
      • Both the set and its elements remain the same
        • Even if the elements match, if the sets are different, it’s not an identity mapping (blu3mo)
          • Defining things is important
      • Here X represents a set, so it should be $i{d}_X$ not $i{d}_x$
• Injective
  • In $X\to Y$,
  • Each x corresponds to a distinct y
  • This varies depending on what X is (Y doesn’t matter) (blu3mo)
• Surjective
  • In $X\to Y$,
  • For every y, there exists a corresponding x
    • This varies depending on what Y is (X doesn’t matter) (blu3mo)
      • $$\begin{gathered} h:R^2\to R \\ (x,y)\mapsto x+y \end{gathered}$$ is not surjective
      • $$\begin{gathered} h:R^2\to R^{+} \\ (x,y)\mapsto x+y \end{gathered}$$ is surjective
  • Given that there exists a y corresponding to every x,
    • It means that every x and y are linked
    • (Duplicates are possible, so it’s not a bijective mapping)
• Bijective
  • Both injective and surjective
  • In other words, each x corresponds to only one y
• - Are there any x such that f(x)≤0? (blu3mo) - If it’s bijective, then there should be an inverse mapping, which can also help determine that (takker) - f(x) is 0 for x=0,1, so it’s not injective
• In a visual representation,
  • Injective:
  • Surjective: There is a point on the graph for every height (y)
• Composite Mapping
  • $g\circ f(x):=g(f(x))$ is similar to a function
  • Importantly, for $f:X\to Y,g:Y\to Z$
    • It means that if the output of f(x) doesn’t match the input type of g, they cannot be composed
    • We didn’t pay much attention to this before because it was mostly $R^1$, but it’s crucial
    • It’s like statically typed programming (blu3mo)
      • Math equals programming (a bit extreme) (takker)
      • This realization is important (takker)
• Inverse Mapping
  • Similar to composite mapping, being aware of matching types is crucial
  • Definition: If $g\circ f=i{d}_X$ and $f\circ g=i{d}_X$, then $f^{-1}:=g$
    • I see
    • An example where only one of them is true is shown here,
      • 
• Theorem: If a mapping is bijective, then an inverse mapping exists- If a function is surjective, does it mean that an inverse mapping does not exist?
  • No, that’s not correct. When considering the inverse mapping of a non-injective function, there might not be a unique x corresponding to y, which would violate the concept mentioned in [Tokyo University 1S1 Basic Mathematical Science: Linear Algebra#624e6a9979e1130000baa336].
• It seems like I’m starting to see the relationship between surjectivity and injectivity (blu3mo).