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Chapter 4: Problem 2

Assume \(U=\mathbb{Z},\) and let $$ \begin{array}{l} A=\\{\ldots,-6,-4,-2,0,2,4,6, \ldots\\}=2 Z \\ B=\\{\ldots,-9,-6,-3,0,3,6,9, \ldots\\} \quad=3 Z \\ C=\\{\ldots,-12,-8,-4,0,4,8,12, \ldots\\}=4 Z . \end{array} $$ Describe the following sets by listing their elements explicitly. $$ \begin{array}{ll} \text { (a) } A \cap B & \text { (b) } C-A \\ \text { (c) } A-B & \text { (d) } A \cap \bar{B} \\ \text { (e) } B-A & \text { (f) } B \cup C \\ \text { (g) }(A \cup B) \cap C & \text { (h) }(A \cup B)-C \end{array} $$

Short Answer

Expert verified
(a) 6Z, (b) {\ldots, -12, -8, 8, 12, \ldots}, (c) {\ldots, -4, -2, 2, 4, \ldots}, (d) {\ldots, -4, -2, 2, 4, \ldots}, (e) {\ldots, -9, -3, 3, 9, \ldots}, (f) {\ldots, -12, -9, -8, -6, -4, -3, 0, 3, 4, 6, 8, 9, 12, \ldots}, (g) C, (h) {\ldots, -9, -6, -3, -2, 2, 3, 6, 9, \ldots}

Step by step solution

01

Find the Intersection A 鈭� B

The intersection of sets, denoted by \( A \cap B \), contains elements that are in both sets \( A \) and \( B \).\( A = \{ \ldots, -4, -2, 0, 2, 4, \ldots \} \), \( B = \{ \ldots, -6, -3, 0, 3, 6, \ldots \} \). The common multiples are \( \{ \ldots, -6, 0, 6, \ldots \} = 6\mathbb{Z} \).
02

Find the Difference C - A

The difference \( C - A \) contains elements that are in C but not in A.\( C = \{ \ldots, -8, -4, 0, 4, 8, \ldots \} \), \( A = \{ \ldots, -6, -4, -2, 0, 2, 4, 6, \ldots \} \), so \( C - A = \{-8, 8, -12, 12, \ldots\} = \{x \,|\, x = 4k, k \in \mathbb{Z}, k eq 0 \} \).
03

Find the Difference A - B

The difference \( A - B \) contains elements in A that are not in B.\( A = \{ \ldots, -4, -2, 0, 2, 4, \ldots \} \), \( B = \{ \ldots, -6, -3, 0, 3, 6, \ldots \} \), so \( A - B = \{\ldots, -4, -2, 2, 4, \ldots \} - \{0\} = \{x \,|\, x = 2k, k \in \mathbb{Z}, k eq 0\} \).
04

Find the Intersection A 鈭� B虆

The complement of B is all elements not in B, \(\bar{B} \), then \( A \cap \bar{B} = A - B \). Hence, use the result from step 3: \( \{\ldots, -4, -2, 2, 4, \ldots\} \).
05

Find the Difference B - A

The difference \( B - A \) contains elements in B that are not in A.\( B = \{ \ldots, -6, -3, 0, 3, 6, \ldots \} \), \( A = \{ \ldots, -4, -2, 0, 2, 4, 6, \ldots \} \), so \( B - A = \{\ldots, -9, -3, 3, 9, \ldots\} \).
06

Find the Union B 鈭� C

The union of two sets contains all elements that are in either set.\( B = \{ \ldots, -9, -6, -3, 0, 3, 6, 9, \ldots \} \), \( C = \{ \ldots, -12, -8, -4, 0, 4, 8, 12, \ldots \} \), so \( B \cup C = \{ \ldots, -12, -9, -8, -6, -4, -3, 0, 3, 4, 6, 8, 9, 12, \ldots \} \).
07

Find the Intersection (A 鈭� B) 鈭� C

Find \( A \cup B\) first, then intersect that with C.\( A \cup B = \{\ldots, -9, -6, -4, -3, -2, 0, 2, 3, 4, 6, 9, \ldots\} \), \( C = \{\ldots, -12, -8, -4, 0, 4, 8, 12, \ldots\} \), so \( (A \cup B) \cap C = \{\ldots, -12, -8, -4, 0, 4, 8, 12, \ldots\} \).
08

Find the Difference (A 鈭� B) - C

Remove elements of C from \(A \cup B\).\( A \cup B = \{\ldots, -9, -6, -4, -3, -2, 0, 2, 3, 4, 6, 9, \ldots \} \), \( C = \{\ldots, -12, -8, -4, 0, 4, 8, 12, \ldots\} \), so \( (A \cup B) - C = \{\ldots, -9, -6, -3, -2, 2, 3, 6, 9, \ldots\} \).

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Key Concepts

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

Intersection of Sets
When dealing with the **intersection of sets**, we are looking for elements that are common to both sets. Imagine two overlapping circles in a Venn diagram. The intersection is the part where they overlap.
For example, if we have sets \( A \) and \( B \), the intersection, \( A \cap B \), includes only those numbers that appear in both sets.
In this exercise, \( A \) consists of multiples of 2 and \( B \) comprises multiples of 3. The common multiples are numbers like -6 and 6 that both sets share, forming \( \{\ldots, -6, 0, 6 \ldots\} \), which simplifies to multiples of 6.
Thus, we write this as \( 6\mathbb{Z} \), representing all multiples of 6.
Union of Sets
The **union of sets** is a set that contains all elements from the given sets, without duplicating any common elements. Picture all of the circles in a Venn diagram being shaded 鈥� this represents the union.
For sets \( B \) and \( C \), we take every number from both sets. \( B \) includes multiples of 3, while \( C \) contains multiples of 4. The union \( B \cup C \) will list numbers like -9, 0, 3, 12, and others from both sets.
It can be represented as \( \{ \ldots, -12, -9, -8, -6, -4, -3, 0, 3, 4, 6, 8, 9, 12, \ldots \} \), containing all unique elements from both \( B \) and \( C \).
Set Difference
The **set difference** between two sets \( C - A \) reflects elements present in \( C \) but not in \( A \). This concept is about separating the elements of one set from another.
Imagine you have \( C \) (multiples of 4) and \( A \) (multiples of 2). If a number is found in both sets, it is removed in the difference. For this example, \( C - A \) results in \( \{ \ldots, -12, -8, 8, 12, \ldots \} \), which excludes all numbers that are common to both.
Essentially, they are multiples of 4 that are not also multiples of 2.
Complement of a Set
The **complement of a set** \( \bar{B} \) in the universal set \( U \) refers to all elements in \( U \) that are not in \( B \). When we find \( A \cap \bar{B} \), we look for elements in \( A \) not present in \( B \).
In this situation, \( \bar{B} \) gives us every integer that isn鈥檛 a multiple of 3. Thus, the set \( A \cap \bar{B} \) includes numbers like 2, 4, -2, -4, etc., which belong to \( A \) but avoid any multiple of 3.
The idea of the complement helps reveal what remains from \( A \) once the elements of \( B \) are subtracted.

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Most popular questions from this chapter

Determine which of the following statements are true, and which are false. Explain! (a) \(\\{a\\} \subseteq\\{a, b, c\\}\) (b) \(\\{a\\} \subseteq\\{\\{a, b\\}, c\\}\) (c) \(\\{a\\} \subseteq \rho(\\{\\{a\\}, b, c\\})\)

Give examples of sets \(A\), \(B\) and \(C\) such that: (a) \(A \subset B\) and \(B \subset C,\) and \(A \notin C\) (b) \(A \in B\) and \(B \in C,\) and \(A \in C\)

Comment on the following statements. Are they syntactically correct? (a) \(x \in A \cap x \in B \equiv x \in A \cap B\) (b) \(x \in A \wedge B \Rightarrow x \in A \cap B\)

Is \([-\infty, \infty]\) a legitimate or correct notation? Explain.

Write each of these sets by listing its elements explicitly (that is, using the roster method). (a) \(\\{n \in \mathbb{Z} \mid-6
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