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2

First, a correction. What you are counting is the number of total orders. You forgot the possibility of indifference. For example, $A \sim B \sim C$ is a perfectly fine preference relation. The correct count of preference relations on $3$ elements should be $13$. Preference relations are also called total preorders or weak orders in mathematics. According to ...

0

We can simplify by assuming that $u$ is linear, in which we can treat $u$ as the total dollar value of the set of items, and $f$ to be the total dollar value plus the count of the items. Or, in other words, $f$ adds $1$ to the dollar value of each item. Would you prefer $1000$ pennies to one diamond worth $\\\$100$?$u(1000 \text { pennies}) = 10$,$u(\...

4

Seems like what you want to do is find the formal math definitions and apply them. upper contour sets are open Find a sequence $(x_n,y_n)$ where all elements are in an upper contour set but the limit of the sequence is not. increase in one component leads to a higher point of utility Show this, i.e. $$x' > x \Rightarrow U(x',y) > U(x,y)$$ and  ...

6

In general, it will not represent the same preferences. There seems to be confusion on what "monotonic transformation" means in this context. It does not have much to do with monotonic preferences. We say that the utility function $v:X\to\mathbb{R}$ is a monotonic transformation of the utility function $u:X\to\mathbb{R}$ if there exists a strictly ...

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