Question

If \(J \neq \varnothing\) and \(J \subseteq I,\) does it follow that \(\bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcap_{\alpha \in J} A_{\alpha} ?\) Explain.

Short answer

Yes, if \(J \subseteq I\), it does follow that \(\bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcap_{\alpha \in J} A_{\alpha}\). This is because every element found in every \(A_\alpha\) for \(\alpha\) in \(J\) must also be found in every \(A_\alpha\) for \(\alpha\) in \(I\), as \(J\) is a subset of \(I\). Therefore, the intersection over \(A_\alpha\) for \(\alpha\) in \(J\) is a subset of the intersection over \(A_\alpha\) for \(\alpha\) in \(I\).

Step-by-step solution

Understanding the Symbols

First of all, it is needed to understand the symbols used. \(J \neq \varnothing\) tells us the set \(J\) is not empty. \(J \subseteq I\) means \(J\) is a subset of \(I\), meaning that every element of \(J\) is also an element of \(I\). The symbol \( \bigcap_{\alpha \in I}\) denotes the intersection of all sets \(A_\alpha\) for every \(\alpha\) in the set \(I\), while \( \bigcap_{\alpha \in J}A_{\alpha}\) denotes the intersection of all sets \(A_\alpha\) for every \(\alpha\) in the set \(J\). So, we're comparing whether the intersection over every set \(A_\alpha\) for \(\alpha\) in \(I\) is a subset of the intersection over every set \(A_\alpha\) for \(\alpha\) in \(J\).

Visualizing the Relationships

Let's visualize this to better understand it. If we have a larger set \(I\), and a smaller set \(J\) is contained within \(I\), how does this relate to the intersections of the sets \(A_\alpha\)? Because every \(\alpha\) present in \(J\) is also present in \(I\) (since \(J\) is a subset of \(I\)), any element that is in every \(A_\alpha\) for \(\alpha\) in \(J\) must also be in every \(A_\alpha\) for \(\alpha\) in \(I\). In other words, the intersection of the sets \(A_\alpha\) over all \(\alpha\) in \(J\) must be a subset of the intersection of the sets \(A_\alpha\) over all \(\alpha\) in \(I\).

Formulating the Conclusion

In conclusion, therefore, if \(J \subseteq I\), that does imply that \(\bigcap_{\alpha \in I} A_{\alpha} \subseteq \bigcap_{\alpha \in J} A_{\alpha}\). This is because any element that is in all sets \(A_\alpha\) in \(I\) must also be in all sets \(A_\alpha\) in \(J\), given that \(J\) is a subset of \(I\).

Key concepts

Subset and Superset

In set theory, the concepts of a subset and a superset are fundamental to understanding how different sets relate to each other. Let's unravel these terms with a simple analogy: imagine a box of colored pencils. If you take a handful of pencils and keep them aside, this handful can be seen as a subset of the original box. Here, every pencil in your handful is also in the box; similarly, in set terms, a subset is a set where every element is also contained in another set, often denoted as \( A \subseteq B \).

Conversely, using the same analogy, the box is a superset of the handful of pencils you've taken out. It encompasses them and probably more. A superset, in set language, is essentially the bigger collection that contains all elements of its subset, and this is denoted as \( B \supseteq A \). It is crucial to note that by definition, a set is always a subset and a superset of itself, which is a nice little self-love concept within mathematics!

Intersection of Sets

The intersection of sets can be likened to a crossroad in a bustling city; it is where different paths meet. When dealing with set theory, intersection refers to the common elements shared among sets. Symbolically, this shared area is represented by \( \cap \).

For example, let's consider two sets: \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \). The intersection of these two sets, noted as \( A \cap B \), would be \( \{2, 3\} \), since these are the numbers that appear in both sets. When we extend this concept to include multiple sets, say all sets labeled \( A_\alpha \) indexed by the elements of another set \( I \), the intersection becomes \( \bigcap_{\alpha \in I} A_\alpha \), which includes only those elements found in all the \( A_\alpha \) sets for every index \( \alpha \) in \( I \).

Set Notation

Set notation is the shorthand language of set theory, allowing us to express complex ideas succinctly. For example, rather than saying 'all the elements that are in both set A and set B', we simply write \( A \cap B \) for the intersection. Understanding this notation is critical for clear communication in mathematics.

Symbols in set notation are more than mere abbreviations; they help visualize and perform operations on sets. The notation \( A \subseteq B \) denotes that A is a subset of B, while \( A \subsetneq B \) means A is a proper subset of B (A is contained within B but they are not equal). An important equal-or-subset notation is \( A \subseteq B \), implying A can be equal to or a subset of B, smoothing out any ambiguities in the hierarchy of sets. In exercises, questions involving notation challenge students to prove relations between sets or to compute operations like intersections, unions, and differences using the elements of the sets.