Cardinality bounds on set algebra

Question

Assume two sets $A$ and $B$. Consider the following set of inequalities:

• $\max \left(\lvert A \rvert, \lvert B\rvert \right) \le \lvert A \cup B\rvert \le \lvert A \rvert + \lvert B \rvert $
• $\lvert A \cap B\rvert \le \min\left(\lvert A \rvert, \lvert B \rvert \right)$

I'm looking for a text reference that discusses and proves them and that may include more cardinality bounds on these and other set operations. I'm particularly interested in both math and CS research that may derive a tighter set of bounds on set unions and intersections.

Any suggestions?

Answer 1

Those inequalites are not hard to prove for a person interested in math. That tighter bounds are not possible can be shown by example.
Also not hard to prove is:

$$|A \cup B| + |A \cap B| = |A| + |B|. $$