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Chapter 5: Problem 26

$$ \text { If } U \text { denotes a universal set, then } U^{c}=\emptyset \text {. } $$

Short Answer

Expert verified
In summary, the complement of the universal set U, denoted as \(U^c\), is the empty set (鈭�). This is because all elements under consideration belong to U, so there cannot be any elements outside of U. Thus, no elements belong to \(U^c\), making it an empty set.

Step by step solution

01

Understanding the terms

First, let's understand the terms involved in the statement: 1. Universal Set (U): It is the set that contains all the elements under consideration. In other words, it is the largest set. 2. Set Complement (A^c): The complement of set A is the set of elements which are not in set A. Denoted as: A^c = {x | x 鈭� U, x 鈭� A} 3. Empty Set (饾湋 or 鈭�): It is a set that contains no elements at all. It is unique because every set has a unique complement.
02

Applying the definitions

Now that we know the terms involved and their definitions, let's apply them in this exercise: We want to find the complement of the universal set U, denoted as U^c. Using the definition of set complement, U^c = {x | x 鈭� U, x 鈭� U}. Recall that U is the universal set containing all elements in consideration, therefore the condition x 鈭� U must be false. In other words, there cannot be any element belonging to U^c.
03

Concluding the result

As U^c does not have any elements, it is an empty set. Therefore, based on our analysis and the definitions of universal set, set complement, and empty set, we can conclude that: U^c = 鈭�, which is the complement of the universal set U is an empty set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Universal Set
The universal set, often denoted by the symbol \( U \), is an essential concept in set theory. It represents the largest collection of elements that we are considering in a given context or problem. For example, if we are discussing the set of all natural numbers, then our universal set would include all numbers from 1 to infinity.
A universal set is especially important because it serves as the reference set when determining the complement of any other set. It encapsulates all possible elements within a particular discussion or mathematical problem.
When working with sets, always identify the universal set first. This way, it will be easier to manage other sets and their relationships. Keep in mind the following:
  • The universal set includes every possible element within the context.
  • It helps in finding set complements and intersections.
  • Knowing the universal set is crucial for accurately defining other sets.
Set Complement
The concept of a set complement, denoted as \( A^c \), refers to the set of elements not in the given set \( A \). To illustrate, consider a universal set \( U \) and a subset \( A \). The complement, \( A^c \), consists of all elements in \( U \) that are not in \( A \).
Here's how you determine a set complement:
  • Identify all elements present in the universal set \( U \).
  • List the elements in set \( A \).
  • The complement \( A^c \) will be those elements found in \( U \) but not in \( A \).
This notion is critical in set operations such as unions and intersections. The set complement establishes what elements are excluded and helps in mapping relationships between sets. It's particularly revealing in our exercise, as the complement of the universal set \( U^c \) leads to the observation that no elements from \( U \) can exist outside of it.
Empty Set
In set theory, the empty set, represented by \( \emptyset \) or \( \phi \), describes a set with no elements. It's a fundamental concept, serving as the identity element for the union operation of sets. The empty set is unique because there is precisely one set that contains nothing.
Understanding the empty set involves:
  • Recognizing that it contains no elements.
  • Knowing that the intersection of any set with itself and the empty set is the empty set.
  • Understanding that the complement of a universal set, as evident in our exercise, results in an empty set since no elements exist outside a universal set.
The empty set is an essential concept for proving various theorems and conducting proofs in mathematics. It forms the basis of fundamental set operations and helps clarify the boundaries of a problem.

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Most popular questions from this chapter

Assume that \(B\) is a Boolean algebra with operations \(+\) and . Give the reasons needed to fill in the blanks in the proofs, but do not use any parts of Theorem \(5.3 .2\) unless they have already been proved. You may use any part of the definition of a Boolean algebra and the results of previous exercises, however. For all \(a\) in \(B, a \cdot a=a\). Proof: Let \(a\) be any element of \(B\). Then $$ \begin{array}{rlr} a & =a \cdot 1 & \frac{(a)}{(b)} \\ & =a \cdot(a+\bar{a}) & \frac{(b)}{(a)} \\ & =(a \cdot a)+(a \cdot \bar{a}) & \frac{(d)}{(d)} \\ & =(a)+0 & \\ & =a \cdot a & \text { (e) } \end{array} $$

Indicate which of the following relationships are true and which are false: a. \(\mathbf{Z}^{+} \subseteq \mathbf{Q}\) b. \(\mathbf{R}^{-} \subseteq \mathbf{Q}\) c. \(\mathbf{Q} \subseteq \mathbf{Z}\) d. \(\mathbf{Z}^{-} \cup \mathbf{Z}^{+}=\mathbf{Z}\) e. \(\mathbf{Z}^{-} \cap \mathbf{Z}^{+}=\emptyset\) f. \(\mathbf{Q} \cap \mathbf{R}=\mathbf{Q}\) g. \(\mathbf{Q} \cup \mathbf{Z}=\mathbf{Q}\) h. \(\mathbf{Z}^{+} \cap \mathbf{R}=\mathbf{Z}^{+}\) i. \(\mathbf{Z} \cup \mathbf{Q}=\mathbf{Z}\)

Use mathematical induction and the following definitions to prove each statement in 35-37. If \(n\) is an integer with \(n \geq 3\) and if \(C_{1}, C_{2}, C_{3}, \ldots, C_{n}\) are any sets, \(C_{1} \cup C_{2} \cup C_{3} \cup \cdots \cup C_{n}=\left(C_{1} \cup C_{2} \cup C_{3} \cup \cdots \cup C_{n-1}\right) \cup C_{n}\), and \(C_{1} \cap C_{2} \cap C_{3} \cap \cdots \cap C_{n}=\left(C_{1} \cap C_{2} \cap C_{3} \cap \cdots \cap C_{n-1}\right) \cap C_{n} .\) (More rigorous versions of the definitions are given in Section 8.4.) Generalized Distributive Law for Sets: For any integer \(n \geq 1\), if \(A\) and \(B_{1}, B_{2}, B_{3}, \ldots, B_{n}\) are any sets, then $$ \begin{aligned} \left(A \cap B_{1}\right) \cup\left(A \cap B_{2}\right) \cup & \cdots \cup\left(A \cap B_{n}\right) \\ &=A \cap\left(B_{1} \cup B_{2} \cup B_{3} \cup \cdots \cup B_{n}\right) \end{aligned} $$

Prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set \(U\). For all sets \(A, B\), and \(C, A-(B-C)=(A-B)-C\).

Let the universal set be the set \(\mathbf{R}\) of all real numbers and let \(A=\\{x \in \mathbf{R} \mid-3 \leq x \leq 0\\}, B=\\{x \in \mathbf{R} \mid-1
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