[求助]求高人指点

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Consider a rational preference relation R. Prove that if u(x)=u(y) implies x~y, u(x)>u(y) implies xPy (which means x is strictly preferred to y), the u(.) is a utility function representing R.

My consideration of this problem:

for any x, if there is and only a unique u(x) exist, that is defined as the reflaction of f:x->u(x), to prove this and we can get the existence of UF representing R.

assuming that for a single x, exist more than one u(x) reflecting in the space U, say u(x) and u(y).

if u(x)=u(y), x only have one reflection in U.

if u(x)not=u(y), we can assume either u(x)>u(y) or the opposite. then from above condition we know that this must implies xPy relation. since u(y) also represent x, follows x~y. Here we generate contradiction from our assumption which means the assumption is wrong.

as a result, for a single x, there is only one u exists in the U space, which shows the function relationship between x and u(x).

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关键词：求高人指点 高人指点 relationship Presenting assumption 求助 指点 高人

设C是给定的消费空间，R是在C上定义的一个理性偏好，本题等价于证明：对于任意x,y∈C，当且仅当yRx，有u(y)≥u(x)。

由理性偏好的完备性，下列情况必居其一：xRy，yRx。

依题设（根据逆否命题等价），若yRx，即xPy不成立，则u(x)>u(y)不成立，即u(y)≥u(x)；同时，若u(y)≥u(x)，则要么yPx，要么y～x，即yRx。故当且仅当yRx，有u(y)≥u(x)。

考虑若xRy的情况同样。

非常感谢bz大人！我实在是基本的数理逻辑没学好，解这类证明最faint了。