# Proving |A U B| Given Disjoint A & B

• chocolatelover

#### chocolatelover

Hi everyone,

Could someone please show me how to prove this?

## Homework Statement

Determine |A U B| in terms of |A| and |B| assuming that A and B are disjoint

## The Attempt at a Solution

I know that A U B must be finite because A and B are disjoint, but besides that I don't know how I would go about proving this.

Could someone please show me how to?

Thank you

There are some intuitive ways to answer this...
(i) The easiest way is to draw a Venn diagram and see what you think the answer might be
(ii) Secondly (and more formally) you could formulate the answer in terms of 'indicator functions'

$$i_X(x)= \begin{cases} 0 & \mbox{if }x \notin X \\ 1 & \mbox{if }x \in X$$

Try the first part and then see if you can do the same via the second

Thank you very much

Regards

Just because A and B are disjoint, does not in any way imply that AuB is finite. Just count the elements (assuming both A and B are finite).

Suppose A= {a}, B= {b}. What is AUB? What is |A|? What is |B|? What is |AUB|?

Suppose A= {a, b, c}, B= {u, v, w, x, y, z}. What is AUB? What is |A|? What is |B|? What is |AUB|?

Do those examples give you any ideas? When you have no idea how to do a general problem, look at simple examples.