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Contrapositive, contradiction and construction: common proof methods

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In this post we will talk about three different, all very common, ways of making proofs: contrapositive, contradiction and construction.

Construction:
(jump to the contrapositive or to the contradiction)
Proofs by construction are probably the proofs that make more sense or are more intuitive in nature. When you prove something by construction, you explicitly build the thing that you described to exist, or give an explicit way of verifying what you described. This is very important because more often than not, mathematics can prove things like "an object X satisfying this and that property exists", but without providing means to find such object.

A good example of a proof by construction is the proof that every function $f: \mathbb{R}\to\mathbb{R}$ can be decomposed into a sum $f(x) = O(x) + E(x)$ where $O(x)$ is an odd function and $E(x)$ is an even function, i.e.

$$\begin{cases}O(-x) = -O(x)\\
E(-x) = E(x)\end{cases}\ \forall x \in \mathbb{R}$$

To prove this statement, we w…