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Math Help - Cardinality if infinite sets

  1. #1
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    Cardinality if infinite sets

    Let A=\{1,2,3\} and B =\{a,b,c\}. Since they are finite sets, itís quite obvious that they have the same number of elements.

    I have read the proof that the infinite sets |(0,1)| = |\mathbb{R}|.
    We know that  (0,1) \subset \mathbb{R}, and I know there is a bijective function between the two sets, but how does one explain that  (0,1) and \mathbb{R} have the same number of elements?
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  2. #2
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    Quote Originally Posted by novice View Post
    I have read the proof that the infinite sets |(0,1)| = |\mathbb{R}|.
    We know that  (0,1) \subset \mathbb{R}, and I know there is a bijective function between the two sets, but how does one explain that  (0,1) and \mathbb{R} have the same number of elements?
    What does it mean ‘to have the same number’?
    Each time I pick a number from (0,1) you can match it with a unique number from \mathbb{R}.
    Likewise you pick any number from \mathbb{R} then I can match it with a unique number from (0,1).
    And the matches are all different because of the bijection.
    You and I have the ‘same number’ of elements.
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  3. #3
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    Quote Originally Posted by Plato View Post
    What does it mean Ďto have the same numberí?
    Each time I pick a number from (0,1) you can match it with a unique number from \mathbb{R}.
    Likewise you pick any number from \mathbb{R} then I can match it with a unique number from (0,1).
    You and I have the Ďsame numberí of elements.
    Let 0,1) \rightarrow \mathbb{R}" alt="f0,1) \rightarrow \mathbb{R}" />. Since there is a bijective function f from (0,1) to \mathbb{R}.

    Let us say f(0.25)=3 and f(0.3)=5. If I remove equal number of elements from both sets, say (0,1)-\{0.25\} and R-\{5\}. Now 0.25 \notin (0,1) and 3 \in \mathbb{R} , and 0.3 \in (0,1) and 5 \notin \mathbb{R}.

    Have we lost the bijective function?

    Can we say  <br />
|(0,1)-\{0.25\}|\not=|\mathbb{R}-\{5\}|<br />
?
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    Quote Originally Posted by novice View Post
    Let 0,1) \rightarrow \mathbb{R}" alt="f0,1) \rightarrow \mathbb{R}" />. Since there is a bijective function f from (0,1) to \mathbb{R}.

    Let us say f(0.25)=3 and f(0.3)=5. If I remove equal number of elements from both sets, say (0,1)-\{0.25\} and R-\{5\}. Now 0.25 \notin (0,1) and 3 \in \mathbb{R} , and 0.3 \in (0,1) and 5 \notin \mathbb{R}.

    Have we lost the bijective function?

    Can we say  <br />
|(0,1)-\{0.25\}|\not=|\mathbb{R}-\{5\}|<br />
?
    I have no idea what any of that says. Much less what your point is.
    If A is any finite subset of (0,1) and B is any finite subset of \mathbb{R} then \left| {\left( {0,1} \right)\backslash A} \right| = \left| {\mathbb{R}\backslash B} \right|
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