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In Kuratowski's definition of n-tuples:

$$(a,b,c,d) = \left\{ {\{ a\} ,\{ a,b\} ,\{ a,b,c\} ,\{ a,b,c,d\} } \right\}$$

This means: $$(a,b,c) = \left\{ {\{ a\} ,\{ a,b\} ,\{ a,b,c\} } \right\}$$ Also, $$(a,b,b,c) = \left\{ {\{ a\} ,\{ a,b\} ,\{ a,b,b\} ,\{ a,b,b,c\} } \right\} = \left\{ {\{ a\} ,\{ a,b\} ,\{ a,b\} ,\{ a,b,c\} } \right\} = \left\{ {\{ a\} ,\{ a,b\} ,\{ a,b,c\} } \right\}$$ Which implies: $$(a,b,c) = (a,b,b,c)$$ What am I missing here?

Thanks in advance!

Camilo Arosemena-Serrato
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Ahmed Ali
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1 Answers1

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I think your definition is wrong. In my experience, Kuratowski's definition of $n$-tuples is defined inductively by $$(a,b) = \{ \{a\}, \{a,b\} \}$$ and, for $n>2$, $$(a_1, a_2, \dots, a_n) = (a_1, (a_2, \dots\, a_n))$$ This definition doesn't give rise to any nonsense.

For instance, $$(a,b,c) = \{ \{a\}, \{a, \underbrace{\{ \{b\}, \{b,c\} \}}_{(b,c)} \} \}$$ whereas $$(a,b,b,c) = \{ \{a\}, \{a, \underbrace{\{\{b\}, \{b, \underbrace{\{ \{b\}, \{b,c\} \}}_{(b,c)} \} \}}_{(b,b,c)} \} \}$$

Clive Newstead
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  • The above aren't "falsehoods," they just have problems for real usage. :) – Thomas Andrews Sep 08 '13 at 23:14
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    I'd say $(1,2,3)=(1,2,2,3)$ is false! Or did I misinterpret your comment? – Clive Newstead Sep 08 '13 at 23:16
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    It's a definition. It isn't a falsehood, it just indicates that you can't use the definition for what you want it to be. A falsehood is a very particular thing in mathematics, and it is bad to confuse it with "intuitive" meanings. In particular, I'd call that a "problem" with the definition, not a falsehood. – Thomas Andrews Sep 08 '13 at 23:17
  • @ThomasAndrews: Fair point. Perhaps I meant "should be false"! – Clive Newstead Sep 08 '13 at 23:19
  • Thanks, it seems that I misunderstood the author's explanation for a definition. Sorry if I bothered anyone with this question. :) – Ahmed Ali Sep 08 '13 at 23:26
  • Great answer. Since we define $(a_1,\ldots,a_n)$ as $(a_1,(a_2,\ldots,a_n))$, this means $A^n=A \times A^{n-1}$ (where $a_i \in A \forall i=1,\ldots,n$. Although I have also seen $(a_1,\ldots,a_n)$ defined the other way as $((a_1,\ldots,a_{n-1}),a_n)$, which says $A^n = A^{n-1} \times A$. Too bad these definitions don't agree with each other. I have posted another question on the issue: http://math.stackexchange.com/questions/964092/the-empty-tuple-or-0-tuple-its-definition-and-properties – EthanAlvaree Jan 16 '15 at 01:32