Question

If a1, a2, ..., an and b1, b2, ..., bn are sets such that aj ⊆ bj for j = 1, 2, 3, ..., n, then which of the following is true?

1) ∪j=1naj ⊆ ∪j=1nbj
2) ∪j=1naj Š‡ ∪j=1nbj
3) ∪j=1naj = ∪j=1nbj
4) None of the above

Answer 1

The correct statement is option 1) ∏j=1naj ⊆ ∏j=1nbj, which means the union of the sets aj is a subset of the union of the sets bj.

Step-by-step explanation:

The student's question is asking to identify the correct statement among the options given when dealing with the union of sets where each individual set aj is a subset of the corresponding set bj, for j ranging from 1 to n.

To determine the correct relationship between the union of all aj sets and the union of all bj sets, let's consider how union and subset relations work. If each aj is a subset of bj, then every element of aj is also an element of bj. When we take the union of all aj sets, this collective set will contain all the elements that are in any of the aj sets.

Since each of these elements is also in the corresponding bj set, the union of all aj sets will be a subset of the union of all bj sets. Therefore, the correct statement is option 1) ∏j=1naj ⊆ ∏j=1nbj, which signifies that the union of the aj sets is included in or equal to the union of the bj sets.