Analysis-Synthesis Reasoning

Question

In french textbook we use reasoning called analysis-Synthesis

Raisonnement par analyse-synthèse since i didn't find any version of it in english either in wikipedia i will write an explanation here

You want to find all the solutions of a certain problem (e.g. all the real numbers $x$ such that $x$ is a square). You first start with what's called the analysis : assume you have a solution of your problem (in our example, assume you have a real $x$ such that $x = y^2$ for some $y\in\mathbb{R}$). Then you're going to show that this solution has got a certain property (in our example, that x is positive). Hopefully this property characterizes all the solutions to your problem and the synthesis is about showing that the objects satisfying the previously mentionned property are solutions to your problem (in our example, every positive $x$ is the square of $\sqrt{x}$ so we're done).

To summarize : show that every solution is of a certain form, then show that every object of this form is a solution.

Answer 1

To summarize : show that every solution is of a certain form, then show that every object of this form is a solution.

In mathematical terminology, this is simply an equivalence, which is the same as the conjunction of both forward and backward implications:

(1) If $X$ is a solution then $X$ satisfies $P$.

(2) If $X$ satisfies $P$ then $X$ is a solution.

And this is not a method but rather the very meaning of finding all solutions to a problem, since it means you want to find a property $P$ such that:

For every object $X$, ( $X$ is a solution if and only if $X$ satisfies $P$ ).

The forward implication (1) ensures that by using $P$ we will 'find' all solutions. The backward implication (2) ensures that by using $P$ we will 'find' only solutions.