Cardinality of Set Union/Examples/3 Arbitrary Sets

Example of Use of Cardinality of Set Union
Let $A_1, A_2, A_3$ be finite sets.

Let:

Then:
 * $26 \le \card {A_1 \cup A_2 \cup A_3} \le 28$

Proof
We have that:

and:

Then:

But as $\card {A_1 \setminus A_2} = 2$, we have that:
 * $\card {\paren {A_1 \cap A_3} \setminus A_2} \le 2$

and so:


 * $11 \le \card {\paren {A_1 \cup A_3} \setminus A_2} \le 13$

and so:
 * $11 + \card {A_2} \le \card {\paren {A_1 \cup A_3} \setminus A_2} + \card {A_2} \le 13 + \card {A_2}$

We have that:
 * $A_1 \cup A_2 \cup A_3 = \paren {\paren {A_1 \cup A_3} \setminus A_2} + \cup A_2$

As $\card {A_2} = 15$ the result follows.