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Math Help - Subtraction of cardinality of a set and its subset

  1. #1
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    Subtraction of cardinality of a set and its subset

    Suppose A is finite and B\subseteq A. Prove that |A\backslash B|=|A|-|B|.

    My attempt: I try to prove it by using induction on B

    Base case: Suppose B has 1 element, then

    |A\backslash B|=|A|-1=|A|-|B|

    I don't think my base case is 'rigorous' enough, it feels there is something missing...

    For induction step:

    Suppose for B having n elements, and B\subseteq A,then |A\backslash B|=|A|-|B|.

    Now let B has n+1 elements, let b\in B be an arbitrary element, then B'=B\backslash \{ b\} has n elements, then

    |A\backslash B'|=|A|-|B'|

    But after that, I have no idea how to proceed at all. Please provide any hint necessary. Thanks in advance.
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  2. #2
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    Re: Subtraction of cardinality of a set and its subset

    Quote Originally Posted by daveclifford View Post
    Suppose A is finite and B\subseteq A. Prove that |A\backslash B|=|A|-|B|.
    For finite sets if $N\cap M=\emptyset$ then $|N\cup M|=|N|+|M|$.

    $A=B\cup (A\setminus B)$. Now apply the above.
    Thanks from daveclifford
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  3. #3
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    Re: Subtraction of cardinality of a set and its subset

    Just curious, where do you study and do you focus your study in set theory?
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