A problem about inductive or dense sets like that, if both A and B are inductive, then are An B and AUB also inductive? If both A and B are dense, then are An B and AUB also dense? Why? Problem 5
Help me understand these two problems.
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Given that the set and are inductive sets.
We know that a set is an inductive set if:
- implies that .
(a).
Consider the set .
Since and are inductive sets, therefore:
and , therefore .
Suppose , therefore either or .
Therefore without loss of generality let .
Since is an inductive set, therefore and hence .
Hence is an inductive set.
Consider the set .
Since and are inductive sets, therefore:
and , therefore .
Suppose , therefore and .
Since is an inductive set and , therefore and hence .
Hence is an inductive set.
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