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Math Help - cardinality

  1. #1
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    cardinality

    if X is any infinite set show that |X| + |\mathbb{N}| = |\mathbb{N}|

    thats it thanks
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  2. #2
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    Quote Originally Posted by jbpellerin View Post
    if X is any infinite set show that |X| + |\mathbb{N}| = |\mathbb{N}|

    thats it thanks
    Not true, with the wording you have X=\mathbb{R} is permitted, and

    |\mathbb{R}| + |\mathbb{N}| = |\mathbb{R}| \ne |\mathbb{N}|

    May be you mean:

    if X is any finite set show that |X| + |\mathbb{N}| = |\mathbb{N}|


    CB
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  3. #3
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    Or if X is an infinite set, |X|+|\mathbb{N} |=|X|
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  4. #4
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    oops I meant X+N = X
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  5. #5
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    Well, that depends on what you know about infinity.

    If you know there exists a B \subset X such that |B|=|\mathbb{N}|, you can find a bijection between B\cup \mathbb{N} and \mathbb{N} (that just means B\cup \mathbb{N} is countable), and you've won.

    Indeed, you get X \equiv (X-B)\cup B \equiv (X-B)\cup \mathbb{N} \equiv (X-B)\cup (B\cup \mathbb{N}) \equiv X\cup \mathbb{N}
    (here \equiv means "equipotent to")
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