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Should a backward or bi-directional arrow be used to link every statement in the process of solving an equation?

It is often difficult to find what implies a statement than what the statement implies. I guess solving simple elementary equations is not that hard is because most operations are reversible.

TFR
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    Solving simple elementary equations is not too hard because they are simple and elementary. We don't know how to solve the vast majority of equations out there. Also I'm not sure what this question really is about, sounds a bit philosophical or maybe linguistic? Or are you asking what arrow represents implication and what "if and only if"? Or are you maybe even asking about the meaning of implication? – freakish Mar 08 '21 at 13:17
  • Well, an easy way for elementary equations to get the other direction is just plugging in values and seeing which ones that satisfied a sufficient condition also satisfied a necessary. You can find potential ones one way, check the other. – Andrew Li Mar 08 '21 at 13:20
  • "since it has the form of p implies q." Not exactly; as you said, it is a problem: to find the truth value of e.g. the formula $\exists x (x+a=0)$ – Mauro ALLEGRANZA Mar 08 '21 at 13:51

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The resolution of an equation is asking for a necessary and sufficient condition, or if you prefer, the explicit enumeration of the solutions.

You can present the steps the way you like, but the conclusion will be like the example below:

$$x^2-3x+2=0\iff x=1\lor x=2.$$

Sometimes you are asked to find some solution. Then you establish a sufficient condition.

$$x=1\implies x^2-3x+2=0.$$

Questions that require a necessary condition seem less frequent. For instance

$$x^2-3x+2=0\implies x\in[1,2].$$

If in the process of resolution you introduce alien solutions, you need to eliminate them after the fact.

E.g.

$$\sqrt x=2\to(\sqrt x)^2=4\to x=4\to x=\pm 2$$

but $\sqrt{-2}$ does not exist.

  • Yeah. I mean each individual solution is a sufficient solution but the compound OR statement made by the individual solutions is a sufficient and necessary solution since when x is one of the values the equation hold and when the equation hold x is one of the values. – TFR Mar 08 '21 at 13:31
  • As an aside, how do you know x is in a certain range as in equation 3, or you just mention it as a random example of a necessary condition. – TFR Mar 08 '21 at 13:40
  • @TFR: just random, but you could see it as obtained from $x^2-3x+2\le0$ by some magic. –  Mar 08 '21 at 13:43