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I am working on the exercise from Terence Tao's Analysis I and I wanted to verify the following proof. I have proven one side of the biimplication and am now stuck on the second part (a hint would be sufficient). Since I am self-studying mathematics I would also be happy to hear your comment on the overall structure and style of the proof.

Exercise 3.4.5 Let $X,Y$ be sets, let $f:X\to Y$. Show that $\forall S\subseteq X: f^{-1} \left(f(S)\right) = S$ if and only if $f$ is injective.

\begin{proof} We first show that $f$ is injective implies $f^{-1} \left(f(S)\right) = S$.

We have already proven the cancellation law for bijective functions: $$ \forall x: g^{-1} \left(g(x)\right) = x \text{ and } g \left(g^{-1}(x)\right) = x$$

We use this fact to prove this proposition. We restrict $f:X \to Y$ to the mapping $f_S$ with the domain $S$ and range $f(S)$ $$f_S: S \to f(S), \quad f_S(x) := f(x)$$ Then $f_S$ is a function and in particular $f_S$ is bijection (Proof: Argue by contradiction. Suppose $f_S$ is not a bijection. Then by negating bijection property $\left(\forall y \in f(S) \quad \exists_{1} x \in S: f_S(x)=y \right)$ we obtain one of the following mutually exclusive cases by trichotomy of order on natural numbers:

  1. $\exists y \in f(S) \quad \exists_{0} x \in S: f_S(x)=y$, a contradiction to the definition of the forward image $f(S):=\{f(x): x\in S\}$
  2. $\exists y \in f(S) \quad \exists_{>1} x \in S: f_S(x)=y$, a contradiction to the injectivity assumption of $f$, since $\exists x,x' \in S: x\neq x'$ and $f_S(x)=f_S(x')$ implies $\exists x,x' \in X: x\neq x'$ and $f(x)=f(x')$.

as desired.) Since $f_S: S \to f(S)$ is bijection, we must have $$f_S^{-1} \left(f_S(S)\right) = S$$ since by Exercise 3.6.3 we have $\forall x \in S: f_S^{-1} \left(f_S(x)\right) = x$ hence $\{ f_S^{-1} \left(f_S(x)\right): x \in S\} = \{x \in S: x \in S\} = S$. It is then left to show that $$f_S^{-1} \left(f_S(S)\right) = f^{-1} \left(f(S)\right)$$ The fact that $f_S(S) = f(S)$ is true since $\forall x \in S: f_S(x) = f(x)$ and injectivity of $f$ together imply $\{f(x) \in Y: x \in S\} = \{f_S(x) \in f(S): x \in S\}$. $$f_S^{-1} \left(f(S)\right) = S$$ The fact that $f_S^{-1} \left(f(S)\right) = f^{-1} \left(f(S)\right)$ is true since $\forall y \in f(S): f_S^{-1}(y) = f^{-1}(y)$ and injectivity of $f$ imply $\{x \in X: f(x) \in f(S)\} = \{x \in S: f_S(x) \in f(S)\}$ $$f^{-1} \left(f(S)\right) = S$$ as desired.

Now we show that $f^{-1} \left(f(S)\right) = S$ implies that $f$ is injective.

proof \end{proof}

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Assume f is injective.
Quickly show A subset $f^{-1}f(A).$ Now assume x in $f^{-1}f(A).$
Thus f(x) in f(A); exists a in A with f(x) = f(a).
Conclude x = a in A. Thus the reverse inclusion.

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