For the question below, use these symbols correspondingly:

Preference relation: ?

Strictly preferred to: ?

Indifference: ~

Is a subset of: ? i.e. A ? B means A is a subset of B

Please bold things like R (real numbers), R+ means positive real numbers, vice versa.

1. [This question shows a utility represents a preference relation if and only if it preserves strict preference and preserves indifference. This is a useful way to think about utility representation, and also gives you a different way of checking utility representation] Let i; be a preference relation on a set X , and let u : X > R. (a) Suppose u is a utility representation for 3. Show that: i. Any :1: > y have u(:1:) > u(y). ii. Any :1: ~ y have u(;r) = u(y). (b) Suppose any :1: > y have u(.1:) > u(y) and any :16 N y have u(:c) = u(y). Show that u is a utility representation for f