. Let X be a set, and let 1 and 2 be two binary relations on...

 

Transcribed Image Text:

. Let X be a set, and let 1 and 2 be two binary relations on X. Define the lexicographic binary relation L as follows: for any x, y X, we have x Lif and only if at least one of the following two conditions holds:. x 1 Y x~1 y and x 2 y. [If 1 and 2 capture how a decision maker feels about different aspects of a decision, then L is one way of combining them: use aspect 1 to make decisions, but if aspect 1 does not make the decision easy, then use aspect 2 to break ties.] Explain why each of the following is true: (a) If x, y X have x Ly, then x 1 y. (b) is complete if both 1 and 2 are complete. (c) is transitive if both 1 and 2 are transitive. [Hint: part (a) of this question and parts (b,c) of the previous question may be helpful.] (d) is a preference relation if both 1 and 2 are preference relations. (e) If 2 is exactly 1, then L is exactly 1. . Let X be a set, and let 1 and 2 be two binary relations on X. Define the lexicographic binary relation L as follows: for any x, y X, we have x Lif and only if at least one of the following two conditions holds:. x 1 Y x~1 y and x 2 y. [If 1 and 2 capture how a decision maker feels about different aspects of a decision, then L is one way of combining them: use aspect 1 to make decisions, but if aspect 1 does not make the decision easy, then use aspect 2 to break ties.] Explain why each of the following is true: (a) If x, y X have x Ly, then x 1 y. (b) is complete if both 1 and 2 are complete. (c) is transitive if both 1 and 2 are transitive. [Hint: part (a) of this question and parts (b,c) of the previous question may be helpful.] (d) is a preference relation if both 1 and 2 are preference relations. (e) If 2 is exactly 1, then L is exactly 1.