# 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 i
nterest 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 dedu
ction, I could avoid complex calculations. I don't understand why
people disregard this method and abduction in general. Sure, it di
dn't prove the uniqueness. To prove the uniqueness, I literally ha
d to solve the problem. But look at the slightly modified next pro
blem.




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 ans
wer 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. The
n 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 uniqu
eness. In fact, even in the first problem I don't think that peopl
e will parametrize if they haven't gotten a solution already. At l
east I would not.




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

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