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  1. #1
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    question on finite sets

    I'm reading through a book on Real Analysis (Intro.) and got a question on finite sets. The definition I'm using for a finite set is: a set A is finite iff there is a one-to-one function f on Nk onto A (Nk is read N sub k where k is a subscript and Nk is a subset of the natural numbers from 1 to k) for some k.

    Question
    Let A = {1, 2, 3, 2}, Nk = {1, 2, 3, 4}, and assume that f is bijective from Nk onto A. Observe that f(2) = f(4) implies 2 ≠ 4. It appears that f is not bijective. Is A finite or not? If A is finite, could you describe a bijective function f from Nk onto A.
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    Quote Originally Posted by klmsurf View Post
    I'm reading through a book on Real Analysis (Intro.) and got a question on finite sets. The definition I'm using for a finite set is: a set A is finite iff there is a one-to-one function f on Nk onto A (Nk is read N sub k where k is a subscript and Nk is a subset of the natural numbers from 1 to k) for some k.

    Question
    Let A = {1, 2, 3, 2}, Nk = {1, 2, 3, 4}, and assume that f is bijective from Nk onto A. Observe that f(2) = f(4) implies 2 ≠ 4. It appears that f is not bijective. Is A finite or not? If A is finite, could you describe a bijective function f from Nk onto A.
    Thanks
    Hi klmsurf.

    Yes, A is finite, but you want to take N_3=\{1,2,3\}, not N_4. Then f is a bijection A\to N_3 with, e.g., 1\mapsto1,\ 2\mapsto2,\ 3\mapsto3.
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    If a bijection exists between two sets, we say they have the same cardinality. A = {1, 2, 3, 2} has exactly four elements, regardless of the distinctness between the elements. Can you explain why the cardinality of A = {1, 2, 3, 2} is 3? Maybe I'm missing a definition or proposition on sets.
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    Quote Originally Posted by klmsurf View Post
    If a bijection exists between two sets, we say they have the same cardinality. A = {1, 2, 3, 2} has exactly four elements, regardless of the distinctness between the elements. Can you explain why the cardinality of A = {1, 2, 3, 2} is 3?
    A = \{1, 2, 3, 2\}= \{1, 2, 3\} it just has an element listed twice.
    B = \{c,c,c,d,c,d\}= \{c,d\} has only two elements.

    May I suggest that you study a foundations of mathematics text before you try Intro to Analysis.
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