For i ∈𝐙, let Ai={i-1, i+1}. Determine the following: (a) ⋃i=1^5 A2 i (b) ⋃i=1^5(Ai ∩Ai+1) (c) ⋃

For i ∈𝐙, let Ai={i-1, i+1}. Determine the following: (a)...

For $i \in \mathbf{Z}$, let $A_i=\{i-1, i+1\}$. Determine the following:
(a) $\bigcup_{i=1}^5 A_{2 i}$
(b) $\bigcup_{i=1}^5\left(A_i \cap A_{i+1}\right)$
(c) $\bigcup_{i=1}^5\left(A_{2 i-1} \cap A_{2 i+1}\right)$.

Key Concepts

Union of Sets

The union operation in set theory combines all elements from the given collections, ensuring that each unique element appears only once in the resulting set. It is symbolized by ? and signifies membership if an element exists in at least one of the considered subsets.

Intersection of Sets

The intersection of sets is a fundamental operation that identifies the common elements shared by all sets involved. Denoted by ?, an element is part of the intersection only if it is present in every one of the sets being intersected.

Indexed Sets

Indexed sets involve a collection of sets each associated with an index, usually drawn from a sequence or another set of indices. Operations such as unions and intersections can be performed across these indexed sets, allowing for a structured and succinct notation to represent the aggregation or commonality of elements across multiple sets.

Transcript

00:01 This question we are required to compute the infinite union and infinite section for some specific set, for sequence of set a i.

00:10 Okay, first, the first a i, the first sequence a i is equal to, okay, minus i, minus i plus 1, blah, blah, blah, minus 1, 0, 1, 2, i.

00:29 Or a i can be written as n is some integers and the absolute of n is less or equal to i.

00:43 Okay, so it's easy for us to see a i, a 1 is a subset of a 2, is a subset of a 3, is a subset of a i, and blah, blah, blah.

00:55 So it's a fact that we can observe.

01:00 And that means the infinite union will be just equal to this one, right? because if we use this expression, we know the infinite union will be actually equal to the union of all sets, such that n is the integer, the absolute value of n is less or equal to i.

01:33 You can see i approaches infinite, and the range of our absolute value also approaches positive infinity.

01:41 So this is actually equal to the integer.

01:46 And once we notice this fact, we find the intersection of a 1 and a 2 is equal to a 1.

01:54 And if we take a 3, we still get a 1.

01:58 So this infinite intersection is just equal to a 1.

02:05 Which is actually equal to minus 1, 0, 1.

02:11 Okay, this is the first sequence.

02:15 The second sequence, a i is equal to minus i and i.

02:23 Then we can see a i and a j are some disjoint set if i is not equal to j.

02:36 Because, for example, a 1 contains minus 1 and 1.

02:40 A 3 contains minus 3 and a 3.

02:42 They are disjoint with each other.