Question

Is \(\bigcap_{\alpha \in I} A_{a} \subseteq \bigcup_{\alpha \in I} A_{\alpha}\) always true for any collection of sets \(A_{\alpha}\) with index set \(I ?\)

Short answer

Yes, \(\bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcup_{\alpha \in I} A_{\alpha}\) is always true for any collection of sets \(A_{\alpha}\) with index set \(I\).

Step-by-step solution

Understand the Problem

It is crucial to first understand that an intersection, denoted as \(\bigcap\), is finding the common elements of each set, while a union, denoted as \(\bigcup\), is combining all unique elements of each set. We are asked if the intersection of all \(A_{\alpha}\) is always a subset of the union of all \(A_{\alpha}\).

Consider an element in the intersection of all \(A_{\alpha}\)

Let's take an arbitrary element, say \(x\), in \(\bigcap_{\alpha \in I} A_{\alpha}\). By definition, this means that \(x\) is in every \(A_{\alpha}\) set where \(\alpha\) belongs to the index set \(I\).

Relation of that element to the union of all \(A_{\alpha}\)

Since \(x\) is in every \(A_{\alpha}\) and a union combines all unique elements of each \(A_{\alpha}\), it follows that \(x\) must be in the union of all \(A_{\alpha}\). This holds true for any element in \(\bigcap_{\alpha \in I} A_{\alpha}\). Therefore, we can conclude \(\bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcup_{\alpha \in I} A_{\alpha}\).

Key concepts

Intersection of Sets

Imagine you have several groups of objects, and you want to know which items they all have in common. That's where the intersection of sets comes into play. In set theory, the intersection is represented by the symbol \( \bigcap \). When you see \( \bigcap_{\alpha \in I} A_{\alpha} \), you're looking at all the elements that are in every single set \( A_{\alpha} \) that's indexed by \( \alpha \) from the index set \( I \).

Here's an analogy—imagine a group of friends deciding on a place to eat. The intersection would represent the restaurants that each and every friend likes. Only the places that all of them agree on will be part of this intersection. Similarly, if an element is in the intersection of a collection of sets, it means that element is found in all the sets in the collection.

Union of Sets

Now, let's talk about the union of sets. While the intersection is focused on commonality, the union is about inclusivity. Symbolized by \( \bigcup \), the union of a collection of sets includes all elements that are in at least one of the sets. In symbols, \( \bigcup_{\alpha \in I} A_{\alpha} \) contains every element that appears in any of the \( A_{\alpha} \) sets indexed by \( \alpha \) from the index set \( I \).

Continuing with the restaurant analogy, the union of friends' preferred eat-out spots would include all the restaurants any of them like. It's a comprehensive list, including every suggestion made by the group, with no repeats. That's the union—it's everyone's preferences rolled into one big list.

Subset Relation

The subset relation is a way to describe when all the elements of one set are also elements of another. This is denoted by \( \subseteq \). If every single item of Set A is found within Set B, then Set A is said to be a subset of Set B. To connect this concept back to the original exercise, when we say \( \bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcup_{\alpha \in I} A_{\alpha} \), we are stating that the intersection of sets (the common elements) is always included within the union of sets (all the elements).

Imagine a puzzle; the pieces that make up the sky portion are completely within the bigger puzzle picture—they're a subset. Just like the puzzle pieces, if an element is part of the intersection (our 'puzzle sky'), then it must be part of the bigger collection—the union.

Index Set

An index set might sound complex, but it's actually a pretty straightforward concept. Think of it as a list of labels or addresses for the sets within a collection. The index set \( I \) is what you use to reference all the individual sets in a family of sets, which are often denoted as \( A_{\alpha} \), \( B_{\beta} \), and so on. The label \( \alpha \) or \( \beta \) is like a house number—it tells you which set you're looking at within the larger neighborhood of sets. When dealing with the intersection and union, the index set \( I \) allows you to identify which sets to intersect or unite.