I have been going through the textbook, "How to do and read proofs" by Daniel Solow. I've been able to understand and implement most concepts correctly, but one concept that was taught early on has continued to plague me throughout the problems sets: The construction method.
I understand when to use the construction method. The issue arises whenever I try to construct the object specified in the conclusion by workings backwards. I have such a misunderstanding of how to construct objects from the conclusion that most of my proof review questions on MSE have revolved around a failure to correctly do so.
Despite reviewing the chapter many times, I continue to fail in constructing the object in a valid way. I am now extremely desperate for help in understanding how to do this correctly.
The Construction Method
When proving that “A implies B” is true, suppose you obtain a statement in the forward process that has the quantifier “there is” in the standard form:
A: There is an “object” with a “certain property” such that “something happens.”
Thus, you can assume that there is such an object, say, X. This object X together with its certain property and the something that happens should help you reach the conclusion that B is true. The technique of working with such an object in the forward process is straightforward and is therefore not given a special name.
In contrast, if you encounter the keywords “there is” during the backward process, then you must show that
B: There is an “object” with a “certain property” such that “something happens.”
One way to do so is to use the construction method. The idea is to construct (guess, produce, devise an algorithm to produce, and so on) the desired object. The constructed object then becomes a new statement in the forward process. However, you should realize that the construction of the object does not, by itself, constitute the proof. Rather, the proof is completed when you have shown that the object you construct is in fact the correct one; that is, that the object has the certain property and satisfies the something that happens, which becomes the next statement in the backward process.
My last MSE question gives an example of the typical failures I experience when attempting to implement the construction method:
A (hypothesis): $a$ and $b$ are real numbers, at least one of which is not $0$, and $i = \sqrt{−1}$.
B (conclusion): There is a unique complex number, say $c + di$, such that $(a + bi)(c + di) = 1$.
B1: $(a + bi)(c + di) = 1$
$\therefore a + bi \not = 0$
B2: $c + di = \dfrac{1}{a + bi}$ where $a + bi \not = 0$
A1: $c + di = \dfrac{1}{a + bi}$ where $a + bi \not = 0$
$\implies (a + bi)(x + yi) = 1$ where $a + bi \not = 0$.
Therefore, in A1, I have constructed the object specified in the conclusion ($c + di$).
I decided to start reading this textbook because I wanted to learn how to do proofs. I'm now almost at the end of the textbook and still have not been able to understand this concept, which was explained in one of the early chapters. It is very disheartening, but I desperately wish to understand it.
I would greatly appreciate it if people could please take the time to explain how to properly/correctly use the construction method to construct the object from the (any) conclusion.
Please note that I am not specifically referring to the example above; I am seeking assistance with regards to the general concept and how to correctly construct the (any) object using this method.
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$\begingroup$Saying $c+id=\frac{1}{a+ib}$ doesn't explain how you "build" $c$ and $d$ from $a$ and $b$. In fact, the "theorem" you are trying to demonstrate is exactly the theorem saying that the notation $\frac{1}{a+ib}$ is meaningful, meaning
- $a+ib$ is invertible,
- the invers is unique.
So with the conclusion, what you have to do is translate it in a system : $$(c+id)(a+ib)=1\iff \left\{\begin{array}{c} ac-bd=1 \\ bc+ad=0 \end{array} \right.$$ and prove this system as a (unique) solution.
The determinant of the system is $a^2+b^2$, so it is $0$ if and only if $a=b=0$. As the hypothesis say $a+ib\ne0$, this can't be.
So the system has a unique solution : $c=\frac{a}{a^2+b^2}$, $d=\frac{-b}{a^2+b^2}$.
You can also discuss the existence of solutions by starting with $bc=-ad$, and discuss weither $b=0$ or $b\ne0$...
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