Prove |X| = |Y| When X\Y and Y\X are Equipotent Sets

[Guest2]

Prove that if X\Y and Y\X are equipotent sets then |X| = |Y|.

The problem is that I've no clue where to start...

(Futile) attempt: There is bijection $f: X\backslash Y \to Y\backslash X$. For every $r_1 \in X\backslash Y$ there exists $r_2$ s.t. $r_2 \in Y\backslash X$. That's $r_1 \in X$ and $r_2 \in Y$. So there's a bijection $f: X \to Y$.

 

[Evgeny.Makarov]

Note that $X=(X\setminus Y)\sqcup (X\cap Y)$ where $\sqcup$ denotes the union of disjoint sets, and similarly $Y=(Y\setminus X)\sqcup (X\cap Y)$.

 

[Guest2]

Note that $X=(X\setminus Y)\sqcup (X\cap Y)$ where $\sqcup$ denotes the union of disjoint sets, and similarly $Y=(Y\setminus X)\sqcup (X\cap Y)$.

I'm not really sure how to use that.

 

[Evgeny.Makarov]

Suppose that you have several married couples with kids. Let $X$ be the set of husbands and children and $Y$ be the set of wives and children. Then, obviously, $X\setminus Y$ (the set of husbands) is in natural one-to-one correspondent with $Y\setminus X$ (the set of wives). Can't you construct a one-to-one correspondence between $X$ and $Y$?