For a real number r, define Ar={r^2}, Br as the closed interval [r-1, r+1] and Cr as the interva

Step 1: Identify the sets u005C( A_u005Calpha u005C), u005C( B_u005Calpha u005C), and u005C( C_u005Calpha u005C) for each u005C(u005Calpha u005C

For a real number r, define Ar={r^2}, Br as the closed...

For a real number $r$, define $A_r=\left\{r^2\right\}, B_r$ as the closed interval $[r-1, r+1]$ and $C_r$ as the interval $(r, \infty)$. For $S=\{1,2,4\}$, determine (a) $\bigcup_{\alpha \in S} A_\alpha$ and $\bigcap_{\alpha \in S} A_\alpha$ (b) $\bigcup_{\alpha \in S} B_\alpha$ and $\bigcap_{\alpha \in S} B_\alpha$ (c) $\bigcup_{\alpha \in S} C_\alpha$ and $\bigcap_{\alpha \in S} C_\alpha$.

For a real number $r$, define $A_r=\left\{r^2\right\}, B_r$ as the closed interval $[r-1, r+1]$ and $C_r$ as the interval $(r, \infty)$. For $S=\{1,2,4\}$, determine
(a) $\bigcup_{\alpha \in S} A_\alpha$ and $\bigcap_{\alpha \in S} A_\alpha$
(b) $\bigcup_{\alpha \in S} B_\alpha$ and $\bigcap_{\alpha \in S} B_\alpha$
(c) $\bigcup_{\alpha \in S} C_\alpha$ and $\bigcap_{\alpha \in S} C_\alpha$.

Key Concepts

Set Operations

Set operations, specifically union and intersection, are methods for constructing new sets from existing ones by combining or finding common elements respectively. In the union of a family of sets, every element that appears in at least one set is included, while the intersection comprises only those elements common to all the sets. This concept is critical when dealing with multiple sets defined by diverse rules or functions.

Singleton Sets

A singleton set is a set that contains exactly one element. In the context of these problems, a singleton set is constructed via a function (like squaring a number), and understanding it is important, as operations like unions and intersections over singleton sets involve comparing individual values rather than intervals or larger collections.

Interval Notation and Properties

Intervals in the real numbers are subsets defined by their endpoints. Closed intervals include their endpoints, while open intervals do not. Recognizing the distinctions between open and closed intervals, as well as how to compute the union and intersection of multiple intervals, is essential for correctly interpreting and solving problems involving ranges of values defined by inequality conditions.

Function Images in Set Definitions

Often sets are defined in terms of a function applied to a parameter (such as squaring a real number or shifting by a fixed amount). Understanding how such function images create sets, and how these sets interact under set operations like union and intersection, is crucial in many areas of mathematics including analysis and algebra.

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