Prove: the cardinality of the set (A-B) is less than or equal to the cardinality of A

Question

Hi this is my first question so please bear with me.

My question is this.

If A and B are sets, is

$ \#(A-B) \leq \#(A) $

True? I drew some Venn diagrams and intuitively this seems to be true,

$ \#(A-B) = \#A $ if the two sets are disjoint and $ \#(A-B) < \#A $ otherwise.

I know that if this is in fact true then

$ \exists $ an injection $ f:(A-B) \to A $

I am having trouble deriving a function $f$ that will convey this.

Answer 1

Simply define

$$f:A\setminus B\to A\;,\;\;f(a):=a$$

The above is an injection and thus $\;\left|A\setminus B\right|\le|A|$.