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I'm having a problem figuring out the correct, or best way to format some equations. I'll give a simple example: Suppose I want to show the identity function $i$ is injective. I need to show the following.

$$i(x_1)=i(x_2) \implies x_1 = x_2 $$

While logically this is clear, I'm not sure how to write it algebraically step by step. Here are some ideas:

$$ \begin{align} i(x_1)=i(x_2) \\ i(x_1)=x_2 \\ x_1 =x_2 \end{align} $$

If this is not ok (and something feels odd about it) then it seems like I'd need multiple equations like:

$$ \begin{align} i(x_1)=i(x_2) \\ =x_2 \end{align} $$ $$ \begin{align} i(x_2)=i(x_1)\\=x_1 \end{align} $$ $$ \therefore i(x_1)=i(x_2) \implies x_2 =x_1 $$

any thoughts?

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    If a function $f$ is such that $f(x_1)=f(x_2) \implies x_1 = x_2$, that does not show that $f$ is surjective. It shows that $f$ is injective. – 2017-02-08
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    sorry, yes typo. I'll change to injective – 2017-02-08

3 Answers 3

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How about $$x_1=i(x_1)=i(x_2)=x_2$$

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    considering that logic flows from top to bottom and left to right, this feels like deducing from what I want to prove. I'm only given $i(x_1)=i(x_2)$, but you're starting with $x_1$. – 2017-02-08
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    The first and last equals are properties of the identity function, and the middle equals is your assumption. The final statement $x_1 = x_2$ follows from transitivity of equality. – 2017-02-08
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    I haven't assumed what I want to prove. The assumption is in the second equality. We can start with the statement $x_1=i(x_1)$ as it is true. – 2017-02-08
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    ok fair enough. I like it. Thanks – 2017-02-08
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To prove $i$ is injective is a "one-liner". By definition, $i(x) = x$ for all $x$, hence
\begin{align*} i(x_1) &= i(x_2)\\[6pt] \implies\; x_1 &= x_2 \end{align*}

Done.

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I'm all for more words.

To prove $i$ is injective I want to show that the only way we can have $i(x) = i(y)$ is for $x$ and $y$ to be the same element of the domain. So

Suppose $i(x) = i(y)$ .

Then [argument here using what you know about the function $i$ in this particular case ...]

So $x$ = $y$.

More compact forms can be correct, of course, and common in professional writing. But beginners are better off wordier. Also I prefer $x$ and $y$ to subscripts $x_1$ and $x_2$, but tastes vary.