Chapter 12: Q 70. (page 932)

Prove that the empty set is both an open subset and a closed subset of $R^{2}$ .

Short Answer

Expert verified

It is proved that the empty set is both an open subset and a closed subset of $R^{2}$ .

Step by step solution

Step 1. Given information.

We have given subset of $R^{2}$ .

Prove the given statement.

The objective is to prove that the given set is both open and closed.

A subset is said to be open if it does not have a boundary to identify.

Consider an empty set. It does not have any boundary to identify. So, it is an open set. The complement of an empty set is the universal set, which again has no boundary. So, universal set is also an open set. Hence, its complement, an empty set, must be a closed set.
Thus, an empty set

\emptyset \subset R^{2}

is Both Open and Closed Set.

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Prove that the empty set is both an open subset and a closed subset of $R^{2}$ .

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

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Prove that the empty set is both an open subset and a closed subset of R2.

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Prove that the empty set is both an open subset and a closed subset of $R^{2}$ .