Question

Prove that transitivity implies each of the following claims: . If a ~ b andb ~~ c, then a c. bc, then da (Harder) If a - bandb~c, then a c. In each case, first convert the strict/indifferent statement to an equivalent statement using only weak preference. Then prove the equivalent statement using our definition of transitivity, which involves only weak preference.

Answer 1

Solution:

First, converting strict preference into weak preference: We know if A is strictly preferred to B (A (curly)> B), this means A is at least as good as B, but B is not at least as good as A (A (curly)>= B, but B is not (curly)>= A). Similarly if a person is indifferent between A and B (A~B), in terms of weak preference we can say that A is at least as good as B and B is at least as good as A (A (curly)>= B and B (curly)>= A). Using these facts, we can write the equivalent statements for the given statements as follows:

(The proofs are also in the image)

If a sim b rightarrow a greaterthansim b and b greaterthansim a and b greaterthansim c rightarrow b greaterthansim c and c greaterthansim b, Then a sim c rightarrow a greaterthansim c and c graeterthansim a So, equivalent statement: If a greaterthansim b, b greaterthansim c, b greaterthansim a and c greaterthansim b, then a greaterthansim c and c greaterthansim a The proof of this is very straight forward: Proof: If a greaterthansim b and b greaterthansim c, then using the principle of transitivity rightarrow a greaterthansim c Similarly, if c greaterthansim b and b greaterthansim a, then again using principle of transitivity rightarrow c greaterthansim a From the above two statements, we have a greaterthansim c and c greaterthansim a Hence, proved (2) if a > b Downarrow a greaterthansim b, b notgreaterthansim a, and b sim c Downarrow b greaterthansim c, c greaterthansim b, then a > c Downarrow a greaterthansim c, c notgreaterthansim a Thus, equivalent statement: If a greaterthansim b, b notgreaterthansim a, b greaterthansim c and c greaterthansim b then