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Chapter 3: Chapter 3

Problem 30

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Show that the complement of the complement of a set is the set itself.

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To show that the complement of the complement of a set is the set itself, we need to prove that for any set A, (A')' = A. Since the complement of a set A, denoted as A', contains all elements not in A, and the complement of A', denoted as (A')', contains all elements not in A', we must show that for any element x, x is in (A')' if and only if x is in A. By proving both directions, we establish that (A')' = A, confirming that the complement of the complement of a set is the set itself.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Understand the complement of a set

The complement of a set A (denoted as A') is the set of all elements that are not in A. In other words, A' = {x | x ∉ A}.

Step 2: Understand the complement of the complement of a set

We are asked to show that (A')' = A. The complement of A' (denoted as (A')') is the set of all elements that are not in A'. In other words, (A'') = {x | x ∉ A'}.

Step 3: Prove the complement of the complement is the original set

To prove that (A')' = A, we must show that for any element x, x is in (A')' if and only if x is in A: 1. If x is in (A')', then by definition, x ∉ A'. This means that x is not in the set of all elements that are not in A. The only possibility left is for x to be in A. 2. If x is in A, then it means x is not in the set of all elements that are not in A (A'). Therefore, x ∉ A'. Since x is not in A', it must be in the set of all elements that are not in A', which is (A''). By proving both directions (if x is in (A')', then x is in A; and if x is in A, then x is in (A')'), we have shown that the complement of the complement of a set is the set itself, or (A')' = A.

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