Families of sets, determination of SDR (system of distinct representatives) 
SDR = System of distinct representatives. Given a finite family of sets $X = \{S_1,\ldots,S_n \}$, a system of distinct representatives, or SDR, for the sets in $X$ is a set of distinct elements $x_1,\ldots,x_n$ with $x_i$ belongs to $S_i$ for $1\le i\le n$.

I need to determine whether the families of sets have an SDR or not.


*

*a) $\{1,2,3\},\{2,3,4\},\{3,4,5\},\{4,5\},\{1,2,5\}$

*b) $\{1,2,4\},\{2,4\},\{2,3\},\{1,2,3\}$

*c) $\{1,2\},\{2,3\},\{1,2,3\},\{2,3,4\},\{1,3\},\{3,4\}$


I've been looking for an example that would explicitly show me what happens and explain step by step what to do to find a SDR. Appreciate any explanations.
 A: "Obviously", for the existence of an SDR it is necessary that $\bigcup X$ has at least $n$ elements and similarly for all sub-families.
This immediately kills case c, where $\bigcup X=\{1,2,3,4\}$ and $n=6$.
Actually finding an SDR where it is possible can be treated as a puzzle game (sudoku for beginners) with trial and error, using as heuristic that one starts with selecting something for smaller sets and avoids the early use of "rare" elements.
For example, with a, one might select $x_4=4$ and is then left with the reduced problem for the family $\{1,2,3\}, \{2,3\},\{3,5\},-,\{1,2,5\}$. Then selecting $x_2=2$ reduces the problem to $\{1,3\},-,\{3,5\},-,\{1,5\}$ and from this e.g. $x_1=1\Rightarrow x_5=5\Rightarrow x_3=3$ or $x_1=3\Rightarrow x_3=5\Rightarrow x_5=1$.
A: I realized that the union does not have to cover all of the treatments (vertices). In Wallis, there is an example,
1.4.1 $ \{12, 145, 12, 123\}$ has a SDR but
1.4.2 $ \{12, 145, 12, 13, 23\}$ has none since
$12 \cup 12 \cup 13 \cup 23$ has 3 elements but the size of the union is 4.
