If $C$ is infinite and $B$ is finite, then $C\setminus B$ is infinite.

Question

If $C$ is infinite and $B$ is finite, then $C\setminus B$ is infinite.

Proof:

Suppose $A=C\setminus B$ is finite. Then since $C$ is infinite, $$C=(C\setminus B)\cup(C\cap B)=A\cup(C\cap B)$$ is also infinite. This implies three separate cases:

1. $A$ is finite and $C\cap B$ is infinite,
2. $A$ is infinite and $C\cap B$ is finite, or
3. $A$ and $C\cap B$ are infinite.

We already asssumed that $A$ was finite, so case (1) applies. Hence $C\cap B$ is infinite and so is $(C\cap B)\cup B=B$, but we assumed B was finite. This is a contradiction and so we conclude our assumption that $A$ was finite is false; $A$ must be infinite. $\square$

I was hoping someone could verify if this proof is valid or not. Thanks in advance!

Answer 1

There is no need for separate cases. Since $B$ is finite and if $A=C\setminus B$ is finite, then $$|A\cup B| = |A|+|B|< \infty$$ so $A\cup B$ is also finite. But $A\cup B = C$ and you are done.

Answer 2

To greedoid's answer, I would add that your proof is correct (if verbose), though you should make sure you can prove that the union of two finite sets is finite.

Also note that you can do this inductively if you can prove that for any infinite set $C$ and any $c \in C$, the set $C \setminus \{c\}$ is infinite.

Answer 3

Yes, your proof is right. But I think that it can be improve.

If $C\setminus B$ is finite, then as $(C\setminus B)\cap B=\emptyset$ you have $$ |C|=|(C\setminus B)\cup B|=|(C\setminus B)|+|B|<\omega $$

But it is a contradiction because, by hypotesis, the set $C$ is infinite