SETS
Let S1, ... , Sn be sets satisfying the following condition: Sj ¡ö Sj+1 for all 1 ¡ i ¡ n - 1. Find S1 ¡û S2 ¡û¡¦ ¡ûSn and S1 ¡ú S2 ¡ú ¡¦ ¡ú Sn
To find the union and intersection of the sets S1, S2, ..., Sn, we need to understand the meanings and operations of union and intersection in the context of sets.
Union (denoted by the symbol "∪") of two sets results in a set that contains all the distinct elements from both sets. In other words, it combines the elements of both sets into one set without repetition.
Intersection (denoted by the symbol "∩") of two sets results in a set that contains only the elements that are common to both sets.
Given that Sj is a subset of Sj+1 for all 1 ≤ j ≤ n-1, we can deduce the following:
1. S1 ∪ S2 ∪ ... ∪ Sn: Since each subsequent set contains all the elements of the previous set and possibly more, the union of all these sets will simply be Sn, the last set in the sequence. So, S1 ∪ S2 ∪ ... ∪ Sn = Sn.
2. S1 ∩ S2 ∩ ... ∩ Sn: To find the intersection of all the sets, we need to identify the elements that are present in every set. Since Sj is a subset of Sj+1, and S1 ⊆ S2 ⊆ ... ⊆ Sn, we can conclude that the intersection of all the sets will simply be S1, the first set. Therefore, S1 ∩ S2 ∩ ... ∩ Sn = S1.
To summarize:
S1 ∪ S2 ∪ ... ∪ Sn = Sn
S1 ∩ S2 ∩ ... ∩ Sn = S1