# This Is Not Wrong

Problem. Let x + 1/y = 1, y + 1/(2z) = 1.
Find z + 1/(2x).

Solution. Let x=1/2. Then y=2, z=-1/2. Now the conditions hold. So
the answer is z + 1/(2x) = -1/2 + 1 = 1/2.


Is the solution unique? Yes. But you don't need to prove if your
interest is only the answer. Anyway, I'll prove it for you.

Let y = 2+t. Then

z = -1/(2+2t)
x = 1 - 1/(2+t) = (1+t)/(2+t)
1/(2x) = (2+t)/(2+2t)

Thus z + 1/(2x) = -1/(2+2t) + (2+t)/(2+2t) = (1+t)/(2+2t) = 1/2.




Let's analysis its logical structure.

This solution uses abduction first.

q
p -> q
--------
Maybe p

In this case,

[x + 1/y = 1, y + 1/(2z) = 1]
[x=1/2, y=2, z=-1/2] -> [x + 1/y = 1, y + 1/(2z) = 1]
------------------------------------------------------
Maybe [x=1/2, y=2, z=-1/2]

Then it uses deduction.

p
p -> q
------
q

[x=1/2, y=2, z=-1/2]
[x=1/2, y=2, z=-1/2] -> [z + 1/(2x) = 1/2]
-------------------------------------------
[z + 1/(2x) = 1/2]


This is the power of abduction. By using abduction instead of
deduction, I could avoid complex calculations. I don't understand
why people disregard this method and abduction in general. Sure,
it didn't prove the uniqueness. To prove the uniqueness, I literally
had to solve the problem. But look at the slightly modified next
problem.




Problem. Let 5x-2y-3z = 0, -3x+4y-z = 0.
Find (x^2 + y^2 + z^2) / (xy+yz+zx).

Solution. Let x=1. Then y=z=1. Now the conditions hold. So the
answer is (x^2 + y^2 + z^2) / (xy+yz+zx) = (1+1+1)/(1+1+1) = 1.

Uniqueness. By some linear algebra, you know (x,y,z) <- Ker A. Then
by the first isomorphism theorem,

R^3 --> R^2
 \       ^
  \      | ~=
   >     |          
R^3 / ker A

Thus ker A ~= R, (x,y,z) = k(1,1,1) = (k,k,k). Therefore
(k^2 + k^2 + k^2) / (k^2 + k^2 + k^2) = 1.


In this case, the solution (1,1,1) is important to prove the
uniqueness. In fact, even in the first problem I don't think that
people will parametrize if they haven't gotten a solution already.
At least I would not.




Long story short, abduction gives you fresh information and
hypotheses. Deduction (and induction for science) is great to
justify, but it doesn't give you fresh information.

Use abduction whenever you can, without hesitant. All you have to
do is justifying later.