If u(x) is an ordinal utility function that represents the (weak) preference relation R, then
(a) any strictly monotonic transformation of u(x) also represents $R$, or
(b) any monotonic transformation of u(x) also represents $R$.
Which is the right proposition, (a) or (b)?
I thought that (b) is the right answer, but when I looked up various online sources I found both definitions, so I'm no longer sure.
I thought (a) cannot be right, because the condition for a monotonic transformation is usually formulated as a conditional: F is a strictly monotonic transformation of u if the following holds: (1) if $u(x)>u(y)$, then $F(u(x))>F(u(y))$. But that doesn't deal with the case (2) $u(x)=u(y)$, which represents xIy. Wouldn't $F(u(x))>F(u(y))$ be compatible with (1) and (2), but represent xPy? Thinking about it, however, it seems that the same as (1) with "greater than or equal" wouldn't do it either. Are the monotonicity conditions formulated as biconditionals? I'm confused.