prove sets cardinality inequality

Question

I need to prove that if $$ A , B $$ are infinite sets and it holds that : $$ |A| > |B| $$ then: $$ |A \backslash B| = |A| $$ I guess I just don't what can I say about the cardinality of |A\B| except that it is less or equal to |A|. Thank you,

Noam

Answer 1

This boils down to showing that if A and B are two infinite sets such that $|X|>|Y|$ then $|X \cup Y| = |X|$ To prove this, notice that $|X \cup Y| \ge |X|$ since the map $f: X \to X \cup Y, f(x) = x$ is injective but not necessarily surjective. To prove that $|X \cup Y| \le |X|$, let $X = S \cup T$ where S and T are disjoint and $|S|= |T|$ since |S|= |X| there exist a bijection k from S to X, and because |T| > |Y| a surjective map m from T to Y is not injective, combining k and m in one function yields a surjective function from $X$ to $X \cup Y$, thus $|X \cup Y| \le |X|$. This shows that $|X \cup Y| = |X|$.

Now to prove that |A\B| can't be less than |A| observe that $B \cup A \setminus B = A$. If $|A \setminus B|$ is less than $|A|$ then the union is less than $|A|$