October 21, 2005 12:18 master Sheet number 24 Page number 8 8 Lecture One By the assumption on Q of a no order effect, for any two alterna-tives x and y , one and only one of the following three answers was received for both Q ( x , y ) and Q ( y , x ) : x ± y , I and y ± x . Thus, the responses to R ( x , y ) and R ( y , x ) are well deﬁned. Next we verify that the response to R that we have constructed with the table is indeed a preference relation (by the second deﬁni-tion). Completeness: In each of the three rows, the answers to at least one of the questions R ( x , y ) and R ( y , x ) is afﬁrmative. Transitivity: Assume that the answers to R ( x , y ) and R ( y , z ) are afﬁrmative. This implies that the answer to Q ( x , y ) is either x ± y or I , and the answer to Q ( y , z ) is either y ± z or I . Transitivity of Q implies that the answer to Q ( x , z ) must be x ± z or I , and therefore the answer to R ( x , z ) must be afﬁrmative. To see that
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Binary relation, Transitivity, Preorder, order effect