$\newcommand{\defeq}{\mathrel{\mathop:}=}$

## 2008/02/26

### Computable Functions are Monotonic

• bottom <= n for all n;
• zero <= zero;
• suc m <= suc n if m <= n.
（Hope it's reflexive, transitive, and anti-symmetric.）所以 bottom 是這個 ordering 之下的最小元素，suc (suc bottom) <= suc (suc (suc (suc zero))) = 4，但 3 和 4 之間沒有次序。Monotonicity of computable functions 說，當 f 是個可計算的自然數函數，我們就有
x <= y implies f x <= f y

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Anonymous2/26/2008 3:53 pm 說：

Josh Ko2/26/2008 3:54 pm 說：

Fall2/26/2008 4:23 pm 說：

Anonymous3/25/2009 4:53 am 說：

Consider the following problem and a possible solution:
A preference relation is said to be weakly monotonic if and only if x  y implies that y  x. Show that if  is transitive, locally nonsatiated and weakly monotonic then it is also monotonic

Solution

Suppose that x >> y.
Define  = min {x1 – y1, …., xL – yL} > 0.
For every z in the consumption set, if z is less than distance  of y, then x >> z.
By local nonsatiation there exists z* such that the distance from z* to y is <  and y  z*.
From the fact that x >> z* and weak monotonicity, z*  x.
Part (iii) of Proposition 1.B.1in the Text is implied by transitivity.
Therefore y  x and so  is monotonic.

do you think this solution answers the Question ？