Question

\(A\) subset \(A\) of a metric space \(R\) is said to be bounded if its diameter \(d(A)\) is finite. Prove that the union of a finite number of bounded sets is bounded.

Short answer

The union of a finite number of bounded sets is bounded, as its diameter is finite.

Step-by-step solution

Understanding the Problem

We are tasked with proving that a finite union of bounded sets within a metric space \(R\) is itself bounded. A set \(A\) is bounded if its diameter \(d(A)\) is finite. The diameter \(d(A)\) of a set \(A\) is defined as the supremum (least upper bound) of the distances between any two points in \(A\).

Identifying Given Information

Let's assume \(A_1, A_2, \ldots, A_n\) are a finite number of bounded sets in a metric space \(R\). Each \(A_i\) has a finite diameter \(d(A_i)\). We know \(d(A_i) = \sup\{d(x, y) \mid x, y \in A_i\}\).

Define the Union of Sets

The union of all these sets is \(A = A_1 \cup A_2 \cup \ldots \cup A_n\). We need to prove that this union \(A\), a subset of \(R\), also has a finite diameter.

Analyzing the Diameter of the Union

The diameter \(d(A)\) of the union \(A\) is defined as \(\sup\{d(x, y) \mid x, y \in A\}\). If \(x, y\in A\), then \(x\in A_i\) and \(y\in A_j\) for some \(i\) and \(j\).

Exploring Possible Distances

If \(i = j\), then both points are in the same set \(A_i\), and \(d(x, y)\leq d(A_i)\), which is finite. If \(i eq j\), then \(x\in A_i\) and \(y\in A_j\), and we consider the individual sets' boundedness to find an overall bound.

Creating an Overall Bound

Since every \(A_i\) is bounded with \(d(A_i) < \infty\), define \(D = \max\{d(A_1), d(A_2), \ldots, d(A_n)\}\). Clearly, \(d(A) \leq D + \text{maximum distance between any two sets}\).

Finalizing the Bound and Conclusion

Combine the finite bounds of individual sets and the distance between them. This finite combination yields a finite \(d(A)\) - thus \(A\) is bounded. Therefore, \(d(A)\), the diameter of the union, is finite since it does not exceed the computable finite bound \(D + \text{maximum inter-set distance}\).

Key concepts

Bounded Sets

In a metric space, the concept of bounded sets is crucial because it helps us understand the limits within which a set is confined. A set is known to be bounded if there is a real number that serves as an upper limit for the distances between any two points within the set. Essentially, this means there is a finite "size" or "reach" to the set.
For a subset \( A \) of a metric space \( R \), this property is signified by its diameter, \( d(A) \), being finite. The diameter captures the greatest distance between any pair of points in the set, defined mathematically as:\[ d(A) = \sup\{d(x, y) \mid x, y \in A\} \]
In plain terms, a bounded set is one that doesn't extend infinitely in any direction within the space it inhabits. Understanding this is crucial when we consider operations involving multiple sets, such as union, because it sets the foundation for further analysis. And indeed, when looking at a union of such sets, these boundedness characteristics have important implications.

Diameter

The diameter is a fundamental metric characteristic of a set within a metric space. It quantitatively expresses the extent or size of a set by measuring the furthest distance between two points that belong to the set. This provides us an understanding of the maximal spread of the set.
Mathematically, the diameter \( d(A) \) is calculated as the supremum of all possible distances \( d(x, y) \) where \( x \) and \( y \) are elements of the set \( A \).
It is important because it helps assert whether a set is bounded. For a bounded set, this supremum, or least upper bound, is finite (i.e., \( d(A) < \infty \)). But for unbounded sets, the diameter would be infinite.
The importance of the diameter is emphasized when working with sets in practical scenarios such as in data analysis or geometry, where understanding the extension of a data cloud or a geometric figure within a space is fundamental.

Union of Sets

The union of sets is a basic operation that combines all elements of two or more sets into a single new set. In mathematical terms, the union of sets \( A_1, A_2, \ldots, A_n \) is expressed as \( A = A_1 \cup A_2 \cup \ldots \cup A_n \).
This operation is especially significant when considering properties such as boundedness.
When dealing with a finite number of bounded sets within a metric space, the union of these sets is an interesting aspect to explore. The challenge is to determine whether such a union remains bounded. By examining the maximal distances within and between these sets, one can establish a collective bound, proving that if each constituent set is bounded, their union is also bounded.

• The individual diameters of sets are considered.
• The operation accounts for potential overlap or non-overlap among sets.

By managing these considerations, one arrives at the understanding that the bounded nature of each set contributes to ensuring the union doesn't become unbounded.

Finite Bounds

Finite bounds refer to the restriction that a particular measurement, such as the diameter of a set within a metric space, is not infinite - it is limited to a specific finite value. These bounds are integral to determining boundedness of individual sets and operations on them, like unions.
When discussing the union of finite bounded sets, the concept of finite bounds is pivotal. This is because:

• For each bounded set \( A_i \) in the union, there exists a finite diameter \( d(A_i) \).
• To determine the boundedness of the union \( A_1 \cup A_2 \cup \ldots \cup A_n \), a combined finite bound is computed. This uses the largest individual diameter along with potential maximum distances between different sets.
• This results in an overall finite bound for the union, \( d(A) \).

This computation ensures that the complexity of combining multiple sets does not lead to an infinite extension. Thus, finite bounds are essential in a variety of applications that require assurance of limited extent or spread across larger constructions built from simpler components.