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Chapter 6: Problem 71

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$ \left(A \cup A^{c}\right)^{c}=\varnothing $$

Short Answer

Expert verified
The statement is true. Since the union of set A and its complement set \(A^c\) contains all possible elements in the universe, the complement of this universal set must be the empty set. Thus, \(\left(A \cup A^{c}\right)^{c}=\varnothing\).

Step by step solution

01

Review basic set operations

Before we proceed with the problem, let's quickly review basic set operations. The union of sets A and B (\(A \cup B\)) consists of all elements found in either set A or set B (or both). For set complements, the complement of set A (\(A^c\)) consists of all elements not found in set A.
02

Express the left-hand side of the equation in terms of set operations

The left-hand side of the equation is given by \(\left(A \cup A^{c}\right)^{c}\). This means we first need to find the union of set A and its complement set \(A^{c}\), then find the complement of the resulting set.
03

Find the union of set A and its complement set \(A^{c}\)

Recall that the union of sets A and \(A^{c}\) consists of all elements found in either set A or set \(A^{c}\) (or both). Since \(A^{c}\) contains all elements not found in set A, the union of A and \(A^{c}\) must contain all possible elements in the universe.
04

Find the complement of the union

Now that we know the union of A and \(A^{c}\) contains all elements in the universe, we need to find the complement of this set. The complement of the universal set is the empty set, as there are no elements left outside the universal set.
05

Check if the left-hand side equals the right-hand side

We have found that the left-hand side simplifies to the empty set, which is equal to the right-hand side of the given statement: \(\varnothing\). Having followed these steps, we can now conclude that the given equation is indeed true: \[\left(A \cup A^{c}\right)^{c}=\varnothing\]

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