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Thread: cardinality

  1. #1
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    cardinality

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

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

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

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

    May be you mean:

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


    CB
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  3. #3
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    Or if $\displaystyle X$ is an infinite set, $\displaystyle |X|+|\mathbb{N} |=|X|$
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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 $\displaystyle B \subset X$ such that $\displaystyle |B|=|\mathbb{N}|$, you can find a bijection between $\displaystyle B\cup \mathbb{N}$ and $\displaystyle \mathbb{N}$ (that just means $\displaystyle B\cup \mathbb{N}$ is countable), and you've won.

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