Thread: Subtraction of cardinality of a set and its subset

1. 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.

2. Re: Subtraction of cardinality of a set and its subset

Originally Posted by daveclifford
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.

3. 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?