Chapter 12: Q. 5 (page 987)

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
* an open subset of $ℝ^{2} o r ℝ^{3} .$

Short Answer

Expert verified

$A s u b s e t S o f ℝ^{2} i s s a i d t o b e o p e n i f f o r e v e r y p o i n t (x, y) \in S t h e r e i s a n o p e n d i s k D s u c h t h a t (x, y) \in D \subseteq S .$

$A s u b s e t S o f ℝ^{3} i s s a i d t o b e o p e n i f f o r e v e r y p o i n t (x, y, z) \in S t h e r e i s a n o p e n b a l l B s u c h t h a t (x, y, z) \in B \subseteq S .$

Step by step solution

Step 1. Given Information.

An open subset of $ℝ^{2} o r ℝ^{3} .$

Step 2. Definition.

$A s u b s e t S o f ℝ^{2} i s s a i d t o b e o p e n i f f o r e v e r y p o i n t (x, y) \in S t h e r e i s a n o p e n d i s k D s u c h t h a t (x, y) \in D \subseteq S .$

$A s u b s e t S o f ℝ^{3} i s s a i d t o b e o p e n i f f o r e v e r y p o i n t (x, y, z) \in S t h e r e i s a n o p e n b a l l B s u c h t h a t (x, y, z) \in B \subseteq S .$

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Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
* an open subset of $ℝ^{2} o r ℝ^{3} .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Definition.

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Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.* an open subset ofℝ2orℝ3.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Definition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Calculus

Applied Mathematics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
* an open subset of $ℝ^{2} o r ℝ^{3} .$