1. Consider the subsets ø (empty set) and R of the real numbers. Prove that both ofthese are open and closed in R.

Name: 1. Consider the subsets ø (empty set) and R of the real numbers. Prov
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Description: 1. Consider the subsets ø (empty set) and R of the real numbers. Prove that both ofthese are open and closed in R.

Answered: Consider the subsets ø (empty set) and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Consider the subsets ø (empty set) and R of the real numbers. Prove that both of
these are open and closed in R.

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Step 1

Consider the set of real numbers $ℝ$ .

We know that a subset A of $ℝ$ is open if for every $x \in A$ , there exists an $ε > 0$ such that $(x - ε, x + ε) \subseteq A$ .

We shall prove that the empty set $\emptyset$ and the whole set $ℝ$ is open using the above definition as follows.

Clearly, $\emptyset$ is a subset of $ℝ$ .

But, the empty set $\emptyset$ has no elements.

So, we can say that for every $x \in \emptyset$ , there exists an $ε > 0$ such that $(x - ε, x + ε) \subseteq \emptyset$ as this is true vacuously because there is no x in $\emptyset$ .

Hence, the empty set $\emptyset$ is open.

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