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Thread: Can an empty set be involved in a 1-1 correspondence?

  1. #1
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    Can an empty set be involved in a 1-1 correspondence?

    Say A is empty.

    Could I do A --> B (1-1) or B --> A (1-1)?

    Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chizum View Post
    Say A is empty.

    Could I do A --> B (1-1) or B --> A (1-1)?

    Thanks.
    Every mapping $\displaystyle f:\varnothing\mapsto X$ is injective. To see this merely note that for the condition that there exists $\displaystyle x_1,x_2\in\text{Dom}(f)$ such that $\displaystyle f(x_1)=f(x_2)$ we would have to say $\displaystyle x_1,x_2\in\varnothing$. See a problem?

    For $\displaystyle f:X\mapsto\varnothing$ it actually reverts to the same thing. For suppose that $\displaystyle X\ne\varnothing$ then for any $\displaystyle x\in X$ we would have that $\displaystyle f(x)$ does not exist? Why? Thus, if $\displaystyle f:X\mapsto\varnothing$ and $\displaystyle f$ is well-defined then by necessity we must have that $\displaystyle X=\varnothing$.
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