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$Determine the qualities of the given set. (Select all that apply.) {(x,y)∣9 < x^(2)+y^(2) < 25} ◻ open ◻connected ◻simply-connected ◻none of the above$

Determine the qualities of the given set. (Select all that apply.) $\newline$ $\left\{(x, y) \mid 9<x^{2}+y^{2}<25\right\}$ $\newline$ $\square$ open $\newline$ $\square$ connected $\newline$ $\square$ simply-connected $\newline$ $\square$ none of the above

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Math Problems

Algebra 2

Conjugate root theorems

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Q. Determine the qualities of the given set. (Select all that apply.)

\newline

\left\{(x, y) \mid 9<x^{2}+y^{2}<25\right\}

\newline

\square

open

\newline

\square

connected

\newline

\square

simply-connected

\newline

\square

none of the above

Identify Nature of Set: Identify the nature of the set based on the inequality $9 < x^2 + y^2 < 25$ . This inequality describes a region between two circles centered at the origin with radii $3$ and $5$ , respectively. The area is not inclusive of the boundaries since it's strictly greater than $9$ and less than $25$ .
Determine Openness: Determine if the set is open. $\newline$ Since the set does not include the boundary points $(x^2 + y^2 = 9$ or $x^2 + y^2 = 25)$ , it is open.
Check for Connectivity: Check if the set is connected. $\newline$ The area between two concentric circles is a single unbroken region, hence it is connected.
Assess Simply-Connected: Assess if the set is simply-connected. $\newline$ A set is simply-connected if every loop in the set can be continuously contracted to a point within the set. The annular region between two circles is not simply-connected because a loop around the inner circle cannot be contracted to a point without leaving the set.