I want to prove that if there is preference relation, induced choice rule of the set is non-empty if the set is finite. I tried to solve this problem by induction. First, I considered a set with only one element. Therefore, the choice rule of the set is the only element of the set. Then I took two-element set and by definition of completeness, I could prove that the two-element set is not empty. And for the third element set, I used the definition of completeness and transitivity to prove that the set is not empty. I generalized this induction for k-1 element set. But I do not know how to connect the $kth$ element to the induction to reach the ultimate proof.
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1I have no idea what this question is about. It reads like a computer generated thing, rather than math. I can only conclude that the tag used is entirely wrong. I hope someone, maybe whomever upvoted, can suggest better tags. – Asaf Karagila Sep 18 '17 at 22:15
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1@AsafKaragila Thanks to your comment. The question is about using binary relation in microeconomics. The binary relation is used in proof of the following proposition: Prove that if $C$ > is a preference relation, then $C >(A)$ is not equal to $∅$; whenever A is finite. I know that the question would be solved by induction. I am sure about my way till $k-1th$ element and I want to know how to mathematically write that the $kth$ element is in the set and satisfies completeness and transitivity of a preference relationship – XX13X882017 Sep 19 '17 at 11:13
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1And it never occurred to you to try the tag [tag:economics] instead? – Asaf Karagila Sep 19 '17 at 11:13
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1@AsafKaragila. Thank you for your comment. I'm almost a new member. Your comment will help me to have better questions and tags in the future, – XX13X882017 Sep 19 '17 at 11:16