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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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    http://www.suitcaseofdreams.net/Reciprocals.htm You can understand inverse is unique by image of link, that is proof. – 2017-02-19
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    @TakahiroWaki My question is very clear. Your response does nothing to explain how to correctly create an object using the construction method. – 2017-02-19
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    [This](http://matheducators.stackexchange.com/questions/8482/commonly-taught-method-divides-by-zero) questions is a perfect example for the construction method. Regarding my answer, point 3 corresponds to "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". – 2017-02-19
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    Terrible question(for understanding), many duplicates, very fundamental question. -1. – 2017-02-19

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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

  1. $a+ib$ is invertible,
  2. 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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    BTW, the "notation" $\sqrt{-1}$ is an awful one : as there is no proper order in $\mathbb C$, you can't "decide" which root of $-1$ (there are 2) is defined by the notation. – 2017-02-19
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    Thanks for the response. I don't understand how your post explains how to correctly construct objects? You said that $c + id = \dfrac{1}{a + in}$ doesn't explain how to build $c$ and $d$. But that is my point: I evidently have a misunderstanding of what it means to construct an object and how to construct an object. This is what I am seeking help with. My proof-writing skills are not good, so I require a very "slow" and detailed explanation of this concept. – 2017-02-19
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    See my extended answer. – 2017-02-19
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    But that doesn't give a general explanation of what it means to correctly "construct an object". This is just the example I gave to show what my misunderstanding of this concept is. My lack of understanding is in correctly implementing the proof methodology; specifically, in constructing an object. The example I gave is just to show what I mean when I say "misunderstanding". – 2017-02-19
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    Problem with construction method is that there are infinitely many ways to use it, some requiring about all the human genius... It gives pointers, not the infinite wisdom :-) – 2017-02-19
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    I understand, but I do not understand what is meant by "constructing an object". In my mind, for the above example, I already constructed $c + di$ by saying $c + di = \dfrac{1}{a + bi}$; but I am told that this is not how to construct an object? So what is the misunderstanding? What does it mean to correctly construct an object? – 2017-02-19
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    @ThePointer The notation $\dfrac 1 w$, almost everywhere, doesn't actually mean a quotient between $1$ and $w$, but rather $\dfrac 1 w$ is another notation for $w^{-1}$. And what is $w^{-1}$? Well, $w^{-1}$ means the unique "number" $\eta$ such that $w\cdot \eta=1$, where $\cdot$ denotes some kind of product and $1$ denotes the neutral element for this product. So when you say $c + di = \dfrac{1}{a + bi}$ you're implicitly using the fact that there exists a unique complex number $z$ such that $z\cdot (a+ib)=1$. What is it that you wanted to prove again? – 2017-02-19
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    @GitGud Ok, this makes sense to me. Thanks for the response. So I need to create the object $z$ now, right? So what does it mean to "create it"? This concept makes no sense to me since it isn't clearly defined. – 2017-02-19
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    @ThePointer Judging by your post history, I'm guessing you've dealt with matrices before. Saying $c + di = \dfrac{1}{a + bi}$ just like that is equivalent to saying $A=\dfrac{1}{\begin{bmatrix}1 & 0\\ 0 & 2 \end{bmatrix}}$ just like that. I hope this example makes things more obvious. – 2017-02-19
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    Answering your last comment now. It means to assume it exists, act as if it exists and try to see what (useful) characteristics it should have. In this particular case you want a complex number $z$ such that $z\cdot (a+ib)=1$. Since $z$ is suppose to be a complex number, you know that $z=c+id$, for some real numbers $c$ and $d$. So you want $(c+id)(a+ib)=1$. Due to the definition of the product of complex numbers, this is equivalent to $(ac-bd)+i(ad+bc)=1$ and due to the definition of complex number, this equates to $c$ and $d$ being such that $ac-bd=1\land ad+bc=0$. Refer to this answer. – 2017-02-19
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    After finding $c$ and $d$, you're supposed to check that indeed $(c+id)(a+ib)=1$. This corresponds to "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". – 2017-02-19
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    @GitGud thank you. Your responses are illuminating. But, in your response, we didn't actually find $c$ or $d$; rather, we found $(ac-bd) $ and $ad + bc$? – 2017-02-19
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    @ThePointer Thank you for the kind words. Yes, I didn't find $c$ or $d$. That's up to you $\ddot \smile$ In any case, I think the link provided by Takahiro might help you with that, (I don't remember if it does and I can't open the link at the moment). Let me know later if you still need help. I gotta go for now. – 2017-02-19
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    @GitGud I will study this further. Thank you for the assistance. – 2017-02-19