Difference between revisions of "Every injection yields a bijection onto its image"

Revision as of 11:43, 26 September 2016

Statement

[ilmath]\newcommand{\f}{\overline{f}}[/ilmath]Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] be sets and suppose [ilmath]f:X\rightarrow Y[/ilmath] is any injective map between them. Then we claim that there is a map:

• [ilmath]\overline{f}:X\rightarrow f(X)[/ilmath] given by [ilmath]\overline{f}:x\mapsto f(x)[/ilmath]- where [ilmath]f(X)[/ilmath] denotes the image of [ilmath]X[/ilmath] under [ilmath]f[/ilmath]^{[Note 1]}

that is a bijection between [ilmath]X[/ilmath] and [ilmath]f(X)[/ilmath] (that is to say, we claim that [ilmath]\f[/ilmath] is a bijection, a bijection is any map that is both injective and surjective)

Proof

We must show that [ilmath]\f:X\rightarrow f(X)[/ilmath] is a bijection (that means [ilmath]\f[/ilmath] is both injective and surjective), even though injectiveness is (even more than surjectiveness) trivial, we do it anyway. That's why this page is marked as first-year friendly

1. Injectivity of [ilmath]\f:X\rightarrow f(X)[/ilmath] given by [ilmath]\f:x\mapsto f(x)[/ilmath]
   • We will take injectivity to mean: [ilmath]\forall x_1,x_2\in X[\f(x_1)=\f(x_2)\implies x_1=x_2][/ilmath] (rather than other equivalent conditions documented on the injection page)
   • Let [ilmath]x_1,x_2\in X[/ilmath] be given
     • Suppose [ilmath]\f(x_1)\ne\f(x_2)[/ilmath]
       • By the nature of logical implication we do not care if the RHS ([ilmath]x_1=x_2[/ilmath]) is true or false.
         • By the principle of excluded middle [ilmath]x_1=x_2[/ilmath] must be true xor false^{[Note 2]}, so we're done in either case.
     • Suppose [ilmath]\f(x_1)=\f(x_2)[/ilmath]
       • In order for the implication to be true, we require that this means [ilmath]x_1=x_2[/ilmath]
       • Notice that [ilmath]\f(x_1)=f(x_1)[/ilmath] and [ilmath]\f(x_2)=f(x_2)[/ilmath] (by definition of [ilmath]\f[/ilmath])
       • As [ilmath]f:X\rightarrow Y[/ilmath] was injective itself, this means:
         • [ilmath]\forall x_1,x_2\in X[\f(x_1)=\f(x_2)\implies x_1=x_2][/ilmath]
       • By hypothesis, [ilmath]\f(x_1)=\f(x_2)[/ilmath] so [ilmath]f(x_1)=f(x_2)[/ilmath]
         • This means that [ilmath]x_1=x_2[/ilmath] by injectivity of [ilmath]f[/ilmath]
       • So [ilmath]\f(x_1)=\f(x_2)\implies f(x_1)=f(x_2)\implies x_1=x_2[/ilmath] (then by transitivity of implies [ilmath]\f(x_1)=\f(x_2)\implies x_1=x_2[/ilmath])
         • We know [ilmath]\f(x_1)=\f(x_2)[/ilmath] to be true, therefore, by definition of implies and by knowing [ilmath]\f(x_1)=\f(x_2)\implies x_1=x_2[/ilmath] we must have:
           • [ilmath]x_1=x_2[/ilmath]
     • This completes the first part of the proof
2. Surjectivity of [ilmath]\f:X\rightarrow f(X)[/ilmath] given by [ilmath]\f:x\mapsto f(x)[/ilmath]
   • To be surjective we require: [ilmath]\forall y\in f(X)\exists x\in X[\f(x)=y][/ilmath]
   • Let [ilmath]y\in f(X)[/ilmath] be given
     • By definition of [ilmath]f(X)[/ilmath] (that is [ilmath]f(X):=\{y\in Y\ \vert\ \exists x\in X[f(x)=y]\}[/ilmath] remember) we see:
       • [ilmath]y\in f(X)\iff\exists x\in X[f(x)=y][/ilmath]
     • Choose [ilmath]x\in X[/ilmath] such that [ilmath]f(x)=y[/ilmath] (which we can do thanks to the above)
       • Now [ilmath]\f(x)=f(x)[/ilmath] by definition of [ilmath]\f[/ilmath] and
         • [ilmath]f(x)=y[/ilmath] by our choice of [ilmath]x[/ilmath]
       • So [ilmath]\f(x)=f(x)=y[/ilmath] or just [ilmath]\f(x)=y[/ilmath]
     • We have shown there exists an [ilmath]x\in X[/ilmath] such that [ilmath]\f(x)=y[/ilmath] for our given [ilmath]y[/ilmath]
   • Since [ilmath]y\in f(X)[/ilmath] was arbitrary we have shown this for all [ilmath]y[/ilmath]
   • That completes the surjectivity part of the proof

TODO: Check, I should have left this blank and marked it as low-hanging fruit....

Thus we have shown that [ilmath]\f:X\rightarrow f(X)[/ilmath] given by [ilmath]\f:x\mapsto f(x)[/ilmath] is a bijection

See also

• Bijection
• Surjection

Notes

1. ↑ Formally:
   • for [ilmath]A\in\mathcal{P}(X)[/ilmath] we define [ilmath]f(A):=\{y\in Y\ \vert\ \exists a\in A[f(a)=y]\}[/ilmath]
   So:
   • [ilmath]f(X):=\{y\in Y\ \vert \exists x\in X[f(x)=y] \}[/ilmath] - the set of all things in [ilmath]Y[/ilmath] that are mapped to by [ilmath]f[/ilmath] for some [ilmath]x\in X[/ilmath]
2. ↑ Xor means "exclusive or", [ilmath]A\text{ XOR }B[/ilmath] means [ilmath]A[/ilmath] or [ilmath]B[/ilmath] is true and [ilmath]A\text{ AND }B[/ilmath] is false. That is to say we have [ilmath]A[/ilmath] or [ilmath]B[/ilmath] but not both

References