Find the Bell Numbers of the following set?
a. I = {i, ii, i, ill, I}
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Find the Bell Numbers of the following set?
a. I = {i, ii, i, ill, I}
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Answer:
Bell numbers represent the number of ways a set can be partitioned.
To find the Bell number of set I = {i, ii, i, ill, I}, we need to find all possible partitions of the set.
1. {i, ii, i, ill, I} —> only one unique element, cannot be partitioned
2. {i, ii, ill, I} —> can be partitioned as {i, ii, ill, I} or {{i}{ii, ill, I}} or {{i, I}{ii, ill}}
3. {i, i, ii, ill, I} —> can be partitioned as {i, ii, ill, I, i} or {{i}{ii, ill, I, i}} or {{i, i}{ii, ill, I}} or {{i, I}{ii, ill, i}} or {{i, ii}{ill, I}}
4. {i, ii, i, ill} —> can be partitioned as {i, ii, ill, i} or {{i}{ii, ill}} or {{i, ii}{ill}}
5. {i, i, ii, I, ill} —> can be partitioned as {i, ii, I, ill, i} or {{i}{ii, I, ill, i}} or {{i, i}{ii, I, ill}} or {{i, I}{ii, ill, i}} or {{i, ii}{ill, I}}
Therefore, the Bell number for set I is 5.