# Thread: If |X|=5, how many non-ordered pairs {A,B} consisting of disjoint subsets exist?

1. ## If |X|=5, how many non-ordered pairs {A,B} consisting of disjoint subsets exist?

Hi, I have the following exercise, but I don't know how to solve it so any help you can give me would be great... Thanks!

It says

Suppose that |X|=5. Find how many non-ordered pairs {A,B} consisting of non-empty disjoint subsets A and B of the set X exist.

2. How many pairs of singletons could you draw from the power set? <single, single>

How many pairs of any one pair and the triple which is disjoint to it could you draw? <pair, triple>

How many distinct pairs of one pair and one of the 3 pairs disjoint to it could you draw? <pair, pair>

How many pairs of any one pair and one of the 3 singletons disjoint to it could you draw? <pair, single>

How many pairs of any triple and one of the 2 singletons disjoint to it? <triple, single>

How many pairs of any quadruple and the singleton disjoint to it? <quadruple, single>

3. @tom@ballooncalculus
Where have you considered the $\displaystyle \emptyset$ in any of your counting?

4. Originally Posted by Plato
@tom@ballooncalculus
Where have you considered the $\displaystyle \emptyset$ in any of your counting?
The question says not to.

Don't you need to do a little more though, considering different possibilities for, e.g., the particular 4-subset?

5. Well, for any of the 5 particular quadruples (each giving a pairing of that quadruple with its disjoint single), no. What different possibilities do you see?

6. Like you said, 5 possibilities.

Maybe it's just the way you interpret what your wrote... I'm reading it as counting for one fixed quadruple instead of all possible ones.