Question

If a sequence of sets \(C_{1}, C_{2}, C_{3}, \ldots\) is such that \(C_{k} \subset C_{k+1}, k=1,2,3, \ldots\) the sequence is said to be a nondecreasing sequence. Give an example of this kind of sequence of sets.

Short answer

An example of a non-decreasing sequence of sets could be the following sequence: \(C_1 = \{1\}, C_2 = \{1, 2\}, C_3 = \{1, 2, 3\}, \ldots\), where the general rule is \(C_k = \{1,2,3, \ldots k\}\).

Step-by-step solution

Define the First Set

Let's start with a simple set as the initial set in the sequence. Let \(C_1\) be the set which includes only one element. For instance, \(C_1 = \{1\}\).

Build the Second Set

Next, we need to create a second set, \(C_2\), that includes the first set. This can be achieved by adding another item to the set. For instance, \(C_2 = \{1, 2\}\). Now, \(C_1\) is a subset of \(C_2\).

Continue Constructing Sets

To show the procedure of set building continues indefinitely, construct a third set, \(C_3\), that includes \(C_2\). You can simply add another item to the set. For example, \(C_3 = \{1, 2, 3\}\). Now, \(C_2\) is a subset of \(C_3\). The same procedure can be applied to construct the next sets in the sequence, \(C_4, C_5, \ldots\). The general rule for this sequence then would be \(C_k = \{1,2,3, \ldots k\}\).

Key concepts

Nondecreasing Sequence of Sets

Understanding the concept of a 'nondecreasing sequence of sets' can greatly simplify your work with infinite set constructions and grasping the nature of subsets. Imagine a collection of sets where each set houses at least as many elements as the set before it. This is precisely what we mean by a nondecreasing sequence.

In more formal terms, for a sequence of sets named \(C_1, C_2, C_3, \ldots\), if every set \(C_k\) is a subset of the next set \(C_{k+1}\), then we declare this sequence to be nondecreasing. This implies that as we move from set \(C_k\) to set \(C_{k+1}\), we either encounter the same elements or more, but never less.

An example of such a sequence could be where \(C_1 = \{a\}\), followed by \(C_2 = \{a, b\}\), and then \(C_3 = \{a, b, c\}\), and so on. In this pattern, every subsequent set contains all the elements of the preceding set, creating a chain where each link contains the entirety of the one before it, possibly with additional elements.

Subset

Let's delve into the term 'subset' to provide a clearer understanding of this foundational concept in set theory. Put simply, a subset is a set where every element also belongs to another, larger set. Think of it as a box within a box. The smaller box (the subset) fits entirely within the larger box without introducing any new items that aren't already present in the big box.

To illustrate this, let's say we have a set \(A \), and we want to define a subset \(B\). If every element in set \(B\) can also be found in set \(A\), then it's accurate to state that \(B \subset A\). This relationship suggests that while set \(A\) may contain additional elements not in set \(B\), set \(B\) cannot have elements outside of those in set \(A\).

For instance, assume \(A = \{1, 2, 3, 4\}\) and \(B = \{2, 3\}\). Here, set \(B\) is a subset of \(A\) because 2 and 3 are elements of set \(A\), fulfilling the subset criteria.

Infinite Set Construction

Infinite set construction is an intriguing area of mathematical statistics and set theory that deals with creating sets that do not have a finite number of elements. These sets stretch out indefinitely, continuing without end. Our previous discussions on nondecreasing sequences and subsets lead us naturally into this concept.

Given the nondecreasing sequence example in which \(C_k = \{1,2,3, \ldots k\}\), as we progress with constructing more sets, we're incrementally approaching the conception of an infinite set. By adding an additional element to each successive set, we are never restricted by a finite boundary; we can always have a set \(C_{k+1}\) with one more element than \(C_k\).

When we talk about infinite sets in the context of sequences, we may also consider what’s known as the 'limit' of the sequence. As \(k\) tends to infinity, the set grows unboundedly. The resulting infinite set, in this exercise, would contain all positive integers, typically denoted by \(\mathbb{N}\), which symbolizes the set of all natural numbers.