Suppose D is a block design based on S1 2 v with blocks B1

Suppose D is a block design based on S={1, 2, .., v} with blocks B1, B2, ...Bb. We define a system of distinct representatives (SDR) for D to be a way of selecting a member xi from each block Bi such that x1, x2, ...are all different. If the design D is to have an SDR, it is necessary that D have at least as many treatments as blocks.

Do there exist System of Distinct Representatives for the following blocks? Why or why not? Show all work.

1) 12, 145, 12, 123

2) 12, 145, 12, 13, 23

Solution

12 , 145 , 12 , 123

Although 12 appears twice, we can choose one of them in each case....

2 from the first block

4 or 5 from the second block

1 from the third block

3 from the fourth block

So, we can go with :

2 , 4 , 1 and 3

OR

1 , 4 , 2 and 3

OR

1 , 5 , 2 and 3

OR

2 , 5 , 1 and 3

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Now, on the second one .....

12, 145, 12, 13, 23

We have two 12\'s... So, one of them we choose 1 and on the other, we choose 2

So, 1 and 2 are definitely accounted for.

But now, look at the fourth and fifth, they are 13 and 23

We are forced to choose 3 from 13 because 1 is already accounted for.

Now, from 23, we can choose nothing as both 2 and 3 are accounted for

So, here we cannot do it.

so one cannot four distinct representatives from these blocks.