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I'm aware that there's a simple proof by contradiction for this, mainly:

Assume $x_1 \neq x_2$. If $f(x_1)=f(x_2)$, then $g(f(x_1))=g(f(x_2))$ which, since $g \circ f$ is injective, implies $x_1=x_2$. By contradiction, $f$ is injective.

I wanted to see if the method of direct proof I used is valid. Could someone just skim over it and alert me if there's any gaps? Here's the proof I came up with:

$$\exists ! x \in D(g \circ f) = A (\forall w \in R(g \circ f) \subseteq C)$$ $$\implies \exists ! y \in R(f) \subseteq B (\forall w \in R(g \circ f) \subseteq C)$$ $$\implies \exists ! x \in D(g \circ f)=D(f)=A (\forall y \in R(f) \subseteq B)$$ $$\implies \exists ! x \in A (\forall y \in R(f) \subseteq B)$$

Hence, $f$ is injective.

Rewritten in words:

Because $f$ is injective, there's a unique $x$ for each $g(f(x))\in C$. Since $g \circ f$ is a function from $R(f)$ onto $C$, this implies there's a unique $y\in R(f)$ for each $g(f(x))\in C$. Since there exists a unique $x$ and unique $y$ for each $g(f(x))\in C$, then there must exist a unique $x$ for each $y \in R(f)$, hence making $f$ injective.

To me, I believe the second implication I made, assuming there exits a unique $x$ for every $y$ just because there existed a unique $y$ for each $w$, seems like a large step -- so I'm really doubting if that's a valid move.

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  • $\begingroup$ This might be a duplicate of math.stackexchange.com/questions/1274914/… $\endgroup$ – jibounet Jan 8 '17 at 14:38
  • $\begingroup$ Can you tell me in words the meaning of $\exists ! x \in D(g \circ f) = A (\forall w \in R(g \circ f) \subseteq C)$? I have a hard time making sense of it. $\endgroup$ – Momo Jan 8 '17 at 15:18
  • $\begingroup$ @Momo I was trying to express how $g \circ f$ was injective by stating that there's only one $x \in A$ such that $g(f(x))=w$ (I think I should have it the other way around actually, that is, $\forall w \dots \exists ! x \dots$), and I mentioned that $R(g \circ f) \subseteq C$ because $g \circ f$ isn't surjective, so it's not necessarily the case that $R(g \circ f) = C$. $\endgroup$ – Drew Jan 8 '17 at 16:37
  • $\begingroup$ @Momo I just edited the original post hopefully clarifying everything $\endgroup$ – Drew Jan 8 '17 at 16:51
  • $\begingroup$ From what I understand, I believe that what you intend to write for the definition of injectivity of $g\circ f$ is $$\forall w\in R(g\circ f)\subseteq C \ \exists! x\in D(g\circ f)=A \text{ such that } g\circ f(x)=w$$ which is quite different from what you actually wrote. $\endgroup$ – Momo Jan 8 '17 at 18:32
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For the direct proof, why not just:

$$f(x)=f(y)\Longrightarrow g(f(x))=g(f(y))\Longrightarrow g\circ f(x)=g\circ f(y)\Longrightarrow x=y$$

So $f$ is injective.

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