Rubinstein2005-page24

Rubinstein2005-page24 - 12:18 master Sheet number 22 Page...

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Unformatted text preview: October 21, 2005 12:18 master Sheet number 22 Page number 6 6 Lecture One By a “legal” response we mean that the respondent ticks exactly one of the boxes in each question. To qualify as preferences a legal response must also satisfy two conditions: 1. The answer to at least one of the questions R ( x , y ) and R ( y , x ) must be Yes. (In particular, the “silly” question R ( x , x ) which appears in the questionnaire must get a Yes response.) 2. For every x , y , z ∈ X , if the answers to the questions R ( x , y ) and R ( y , z ) are Yes, then so is the answer to the question R ( x , z ) . We identify a response to this questionnaire with the binary rela- tion on the set X defined by x y if the answer to the question R ( x , y ) is Yes. ( Reminder : An n-ary relation on X is a subset of X n . Examples: “Being a parent of” is a binary relation on the set of human beings; “being a hat” is an unary relation on the set of objects; “ x + y = z ” is a 3-nary relation on the set of numbers; “...
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