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Hypothetically, could you just reverse the steps from the forward direction of the proof to do the backward direction?

For example, let's say in the forward direction, you take some value $x$ and take the square root to get $\sqrt{x}$, and then add something else to it to get $y$.

Could the backward direction be done by subtracting something from $y$, and then squaring that value to get $x$?

amycandyit
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    From the [tag:proof-writing] tag description: "This tag should not be the only tag for a question". – Shaun Nov 08 '21 at 19:22
  • No, not always. Often one direction is easy to prove and the other direction is hard. – littleO Nov 08 '21 at 20:34

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The problem is that not all operations can be reversed. Consider this example:

Let $x \in \mathbb{C}$. Then consider these 2 statements,

  1. $x$ is a real number $\implies$ $x^2$ is a real number

This is true since the real numbers are closed under multiplication, so $x^2 = xx$ is a real number.

  1. $x^2$ is a real number $\implies$ $x$ is a real number

This is false. For example, $x^2 = -1$ is a real number but there is no real number $x$ such that $x^2 = -1$. The reason you can't do the above proof backward is because $f(x) = x^2$ has no inverse function over $\mathbb{R}$.

If all your operations are reversible then yes you can do exactly what you suggested.

Brian Lai
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