That set has 4 elements, so it has 24 or 16 subsets.

There is 1 subset with 0 elements:

 1. { } or ∅

There are 4 subsets with 1 element:

 2. {(1,2)}
 3. {(2,3))
 4. {(3,4)}
 5. {(4,5)}

There are 6 subsets with 2 elements:

 6. {(1,2),(2,3)}
 7. {(1,2),(3,4)}
 8. {(1,2),(4,5)}
 9. {(2,3),(3,4)}
10. {(2,3),(4,5)}
11. {(3,4),(4,5)}

There are 4 subsets with 3 elements:

12. {(1,2),(2,3),(3,4)}
13. {(1,2),(2,3),(4,5)}
14. {(1,2),(3,4),(4,5)}
15. {(2,3),(3,4),(4,5)}

There is 1 subset with 4 elements

16. {(1,2),(2,3),(3,4),(4,5)}

[The last one is the whole set.  Although it is considered a subset
and should be listed as one, it could not be properly called "sub-", 
so we say "It is a subset, but NOT a PROPER subset".]

Edwin