3. Let X be a set, and let ₁ and 2 be two binary relations on X. Define the lexicographic binary relation as follows: for any z, y € X, we have x L if and only if at least one of the following two conditions holds:¹. • 1 y • ï ~₁ y and ï ₂ y. [If 1 and 2 capture how a decision maker feels about different aspects of a decision, then 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 L y, then r Xı y. (b) (c) is complete if both ₁ and ₂ are complete. 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) Ifa is exactly ₁, then is exactly ₁.
3. Let X be a set, and let ₁ and 2 be two binary relations on X. Define the lexicographic binary relation as follows: for any z, y € X, we have x L if and only if at least one of the following two conditions holds:¹. • 1 y • ï ~₁ y and ï ₂ y. [If 1 and 2 capture how a decision maker feels about different aspects of a decision, then 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 L y, then r Xı y. (b) (c) is complete if both ₁ and ₂ are complete. 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) Ifa is exactly ₁, then is exactly ₁.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Let X be a set, and let 1 and 2 be two binary relations on X. Define the
lexicographic binary relation as follows: for any z, y € X, we have x ɩ if
and only if at least one of the following two conditions holds:¹.
• ï >1 Y
• ~₁ y and r₂ y.
[If 1 and 2 capture how a decision maker feels about different aspects of a
decision, then 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 r L y, then r i y.
(b) is complete if both ₁ and ₂ are complete.
AL
(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.]
is a preference relation if both ₁ and 2 are preference relations.
2 is exactly 1, then is exactly ı.
L
(d)
(e) If
(f) If 2 is exactly 1 (see question 2(a)), then is exactly ₁.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd86cda1-af2d-4585-b012-7ab18b7ef9b3%2Fe964b29b-0c89-452d-a454-b37f2b35b3b8%2Fkqs2zb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let X be a set, and let 1 and 2 be two binary relations on X. Define the
lexicographic binary relation as follows: for any z, y € X, we have x ɩ if
and only if at least one of the following two conditions holds:¹.
• ï >1 Y
• ~₁ y and r₂ y.
[If 1 and 2 capture how a decision maker feels about different aspects of a
decision, then 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 r L y, then r i y.
(b) is complete if both ₁ and ₂ are complete.
AL
(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.]
is a preference relation if both ₁ and 2 are preference relations.
2 is exactly 1, then is exactly ı.
L
(d)
(e) If
(f) If 2 is exactly 1 (see question 2(a)), then is exactly ₁.
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