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Chapter 5: Problem 8

For all sets \(A, B\), and \(C\), $$ (A-B) \cup(C-B)=(A \cup C)-B $$

Short Answer

Expert verified
To prove that \((A-B) \cup(C-B)=(A \cup C)-B\), we rewrote the equation in set-builder notation. We demonstrated that any element \(x\) in the left-hand set belongs to the right-hand set, and vice versa. Since both sides contained the same elements, we concluded that the sets are equal, thereby proving the given equality.

Step by step solution

01

Rewrite the problem into set-builder notation

To better understand this exercise, we will first rewrite the given equation using set-builder notation: $$ (A-B) \cup(C-B) = (A \cup C)-B $$ can be written as: $$ \{x : (x \in A \text{ and } x \notin B) \text{ or } (x \in C \text{ and } x \notin B)\} = \{x: (x \in A \text{ or } x \in C) \text{ and } x \notin B \} $$
02

Demonstrate that each element in the left set is in the right set

Let x be an element of the left-hand side of the equation, i.e., $$ x \in (A-B) \cup(C-B) $$ According to the definition of union operation, x must either belong to (A-B) or (C-B). In other words, x must satisfy one of the following conditions: 1. x 鈭� A and x 鈭� B 2. x 鈭� C and x 鈭� B We notice that either condition implies also that x 鈭� B. Now, if x 鈭� A, then x 鈭� A 鈭� C (since A is a subset of A 鈭� C). If x 鈭� C, then, again, x 鈭� A 鈭� C (since C is a subset of A 鈭� C). Therefore, if x 鈭� A and x 鈭� B, or x 鈭� C and x 鈭� B, then x 鈭� (A 鈭� C) and x 鈭� B. This means that x 鈭� (A 鈭� C) - B, which is the right-hand side of the equation.
03

Demonstrate that each element in the right set is in the left set

Now we show that an element in the right set also belongs to the left set. Let x be an element of the right-hand side: $$ x \in (A \cup C)-B $$ This means that x 鈭� A 鈭� C and x 鈭� B. We can break this condition into two cases: 1. x 鈭� A and x 鈭� B: In this case, x 鈭� (A-B), so x 鈭� (A-B) 鈭� (C-B) 2. x 鈭� C and x 鈭� B: In this case, x 鈭� (C-B), so x 鈭� (A-B) 鈭� (C-B) In either case, x is an element of the left-hand side of the equation.
04

Conclusion

Since we have shown that every element in the left-hand set is in the right-hand set, and vice versa, the two sets (A-B) 鈭� (C-B) and (A 鈭� C)-B are equal. We have therefore proved that for any sets A, B, and C: $$ (A-B) \cup(C-B)=(A \cup C)-B $$

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Key Concepts

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

Set Operations
Set operations are fundamental actions we perform with sets to form new sets or to get more information from existing ones. These operations usually involve two or more sets and result in another set. Common set operations include union, intersection, and difference. Understanding these operations allows us to manipulate sets and solve problems in set theory effectively.

- **Union**: Combines all elements from involved sets, whereas, - **Intersection**: Only contains elements common to all sets. - **Difference**: Removes elements of one set from another. Set operations are often used with Venn diagrams to visually display relationships between sets and better understand these concepts. By practicing these operations, students can develop deeper mathematical intuition and solve complex set-related problems.
Set-Builder Notation
Set-builder notation is a concise way of describing a set by specifying a property that its members must satisfy. This notation is incredibly useful because it allows mathematicians to easily express complex sets without listing every element.

Consider this example:\[A = \{ x : x \text{ is an integer and } x > 0 \text{ and } x \leq 10 \}\]This defines set A as containing all integers greater than 0 and less than or equal to 10. When writing sets in this form, it鈥檚 essential to start with \( \{ x : \ldots \} \), which can be read as "the set of all x such that..." followed by the property condition.

This notation helps when dealing with infinite sets or when we cannot list all elements explicitly. Understanding set-builder notation is key to working with advanced mathematical concepts, as it encourages precise and clear communication.
Union and Difference of Sets
The union and difference of sets are specific set operations used frequently in mathematics to understand the relationship between different groups of elements. Let's break these terms down further.
- **Union** (\(A \cup B\)): The union of two sets A and B is a set containing all elements that are in A, or in B, or in both.- **Difference** (\(A - B\)): The difference of two sets A and B, also known as the relative complement, is a set containing elements in A that are not in B.Using these concepts, we can rewrite complicated expressions into simpler, more manageable forms using the set-builder notation as shown in the original exercise. The distributive property is also crucial to understanding how these operations can be manipulated to prove equalities, like in the given equation of the exercise. Mastering these operations enriches one's analytical skills in logical reasoning and problem-solving.

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Most popular questions from this chapter

Assume that \(B\) is a Boolean algebra with operations \(+\) and . Prove each statement without using 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 \(x, y\), and \(z\) in \(B\), if \(x+y=x+z\) and \(x \cdot y=x \cdot z\), then \(y=z\).

$$ \text { For all sets } A \text { and } B \text {, if } A \subseteq B \text { then } B^{c} \subseteq A^{c} \text {. } $$

For all sets \(A, B, C\), and \(D\), if \(A \cap C=\emptyset\) then \((A \times B) \cap(C \times D)=\emptyset\).

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 $$ \begin{aligned} \left(A \cap B_{1}\right) \cup\left(A \cap B_{2}\right) \cup & \cdots \cup\left(A \cap B_{n}\right) \\ &=A \cap\left(B_{1} \cup B_{2} \cup B_{3} \cup \cdots \cup B_{n}\right) \end{aligned} $$

Can there exist a computer program that has as output a list of all the computer programs that do not list themselves in their output? Explain your answer.
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