If The Set $U={$ All Positive Integers $}$[/tex] And Set $A={x \mid X \in U \text{ And } X \text{ Is An Odd Positive Integer} }$, Which Describes The Complement Of Set $ A A A $, A C A^c A C $?A. $A^c={x

Introduction

In mathematics, set theory is a fundamental branch that deals with the study of sets, which are collections of unique objects. Sets can be described using various mathematical operations, such as union, intersection, and complement. In this article, we will focus on understanding the concept of set complement, specifically for the given sets $U$ and $A$. We will explore the definition of set complement, provide examples, and discuss the properties of set complement.

What is Set Complement?

The set complement of a set $A$, denoted as $A^c$, is the set of all elements that are not in $A$. In other words, it is the set of elements that are in the universal set $U$ but not in $A$. The set complement is an essential concept in set theory, as it helps us understand the relationship between sets and their elements.

Universal Set $U$

The universal set $U$ is the set of all positive integers. It is denoted as:

$$U=\{1, 2, 3, 4, 5, \ldots\}$$

Set $A$

The set $A$ is defined as the set of all odd positive integers. It is denoted as:

$$A=\{x \mid x \in U \text{ and } x \text{ is an odd positive integer} \}$$

$$A=\{1, 3, 5, 7, 9, \ldots\}$$

Complement of Set $A$

The complement of set $A$, denoted as $A^c$, is the set of all elements that are not in $A$. Since $A$ is the set of all odd positive integers, the complement of $A$ will be the set of all even positive integers.

$$A^c=\{x \mid x \in U \text{ and } x \text{ is an even positive integer} \}$$

$$A^c=\{2, 4, 6, 8, 10, \ldots\}$$

Properties of Set Complement

The set complement has several important properties that are essential to understand:

• Complement of the Universal Set: The complement of the universal set $U$ is the empty set $\emptyset$, since there are no elements outside of $U$.
• Complement of the Empty Set: The complement of the empty set $\emptyset$ is the universal set $U$, since all elements are in $U$.
• Complement of the Complement: The complement of the complement of a set $A$ is the set $A$ itself, i.e., $(A^c)^c=A$.
• Complement of the Union: The complement of the union of two sets $A$ and $B$ is the intersection of their complements, i.e., $(A \cup B)^c=A^c \cap B^c$.
• Complement of the Intersection: The complement of the intersection of two sets $A$ and $B$ is the union of their complements, i.e., $(A \cap B)^c=A^c \cup B^c$.

Conclusion

In conclusion, the set complement is an essential concept in set theory that helps us understand the relationship between sets and their elements. The complement of a set $A$ is the set of all elements that are not in $A$. We have discussed the universal set $U$, set $A$, and the complement of set $A$. We have also explored the properties of set complement, including the complement of the universal set, the complement of the empty set, the complement of the complement, the complement of the union, and the complement of the intersection.

Example Problems

Here are some example problems to help you practice your understanding of set complement:

1. Find the complement of the set $A=\{1, 2, 3, 4, 5\}$.
2. Find the complement of the set $B=\{x \mid x \in U \text{ and } x \text{ is a multiple of 3} \}$.
3. Find the complement of the union of the sets $A$ and $B$.
4. Find the complement of the intersection of the sets $A$ and $B$.

Solutions

1. The complement of the set $A=\{1, 2, 3, 4, 5\}$ is the set of all elements that are not in $A$. Since $A$ is a subset of the universal set $U$, the complement of $A$ will be the set of all elements in $U$ but not in $A$.

$$A^c=\{6, 7, 8, 9, 10, \ldots\}$$

2. The set $B$ is defined as the set of all multiples of 3. The complement of $B$ will be the set of all elements that are not multiples of 3.

$$B^c=\{x \mid x \in U \text{ and } x \text{ is not a multiple of 3} \}$$

$$B^c=\{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, \ldots\}$$

3. The union of the sets $A$ and $B$ is the set of all elements that are in $A$ or in $B$.

$$A \cup B=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \ldots\}$$

The complement of the union of the sets $A$ and $B$ is the set of all elements that are not in $A$ or in $B$.

$$(A \cup B)^c=\emptyset$$

4. The intersection of the sets $A$ and $B$ is the set of all elements that are in both $A$ and $B$.

$$A \cap B=\{3, 6, 9, 12, \ldots\}$$

The complement of the intersection of the sets $A$ and $B$ is the set of all elements that are not in both $A$ and $B$.

(A \cap B)^c=\{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, \ldots\}$<br/> **Set Complement Q&A** =====================

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about set complement.

Q: What is the set complement?

A: The set complement of a set $A$, denoted as $A^c$, is the set of all elements that are not in $A$. In other words, it is the set of elements that are in the universal set $U$ but not in $A$.

Q: How do I find the complement of a set?

A: To find the complement of a set $A$, you need to identify the elements that are not in $A$. This can be done by listing all the elements in the universal set $U$ and then removing the elements that are in $A$.

Q: What is the difference between the complement and the union of two sets?

A: The complement of two sets $A$ and $B$ is the set of all elements that are not in both $A$ and $B$. The union of two sets $A$ and $B$ is the set of all elements that are in $A$ or in $B$.

Q: Can the complement of a set be empty?

A: Yes, the complement of a set can be empty. This occurs when the set is the universal set $U$, in which case the complement is the empty set $\emptyset$.

Q: Can the complement of a set be the universal set?

A: No, the complement of a set cannot be the universal set. This is because the complement of a set is the set of all elements that are not in the set, whereas the universal set contains all elements.

Q: How do I find the complement of the union of two sets?

A: To find the complement of the union of two sets $A$ and $B$, you need to find the intersection of their complements, i.e., $(A \cup B)^c=A^c \cap B^c$.

Q: How do I find the complement of the intersection of two sets?

A: To find the complement of the intersection of two sets $A$ and $B$, you need to find the union of their complements, i.e., $(A \cap B)^c=A^c \cup B^c$.

Q: Can the complement of a set be equal to the set itself?

A: No, the complement of a set cannot be equal to the set itself. This is because the complement of a set is the set of all elements that are not in the set, whereas the set itself contains all elements.

Q: How do I use set complement in real-world applications?

A: Set complement is used in various real-world applications, such as:

• Data analysis: Set complement is used to identify the elements that are not in a particular dataset.
• Machine learning: Set complement is used to identify the elements that are not in a particular model.
• Computer science: Set complement is used to identify the elements that are not in a particular program.

Conclusion

In conclusion, set complement is an essential concept in set theory that helps us understand the relationship between sets and their elements. We have discussed the definition of set complement, provided examples, and addressed some of the most frequently asked questions about set complement. We hope that this article has helped you understand the concept of set complement and its applications.

Example Problems

Here are some example problems to help you practice your understanding of set complement:

1. Find the complement of the set $A=\{1, 2, 3, 4, 5\}$.
2. Find the complement of the set $B=\{x \mid x \in U \text{ and } x \text{ is a multiple of 3} \}$.
3. Find the complement of the union of the sets $A$ and $B$.
4. Find the complement of the intersection of the sets $A$ and $B$.

Solutions

1. The complement of the set $A=\{1, 2, 3, 4, 5\}$ is the set of all elements that are not in $A$. Since $A$ is a subset of the universal set $U$, the complement of $A$ will be the set of all elements in $U$ but not in $A$.

$$A^c=\{6, 7, 8, 9, 10, \ldots\} </span></p> <ol start="2"> <li>The set <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is defined as the set of all multiples of 3. The complement of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> will be the set of all elements that are not multiples of 3.</li> </ol> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>B</mi><mi>c</mi></msup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∣</mo><mi>x</mi><mo>∈</mo><mi>U</mi><mtext> and </mtext><mi>x</mi><mtext> is not a multiple of 3</mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B^c=\{x \mid x \in U \text{ and } x \text{ is not a multiple of 3} \} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mord text"><span class="mord"> and </span></span><span class="mord mathnormal">x</span><span class="mord text"><span class="mord"> is not a multiple of 3</span></span><span class="mclose">}</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>B</mi><mi>c</mi></msup><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo separator="true">,</mo><mn>7</mn><mo separator="true">,</mo><mn>8</mn><mo separator="true">,</mo><mn>10</mn><mo separator="true">,</mo><mn>11</mn><mo separator="true">,</mo><mn>13</mn><mo separator="true">,</mo><mn>14</mn><mo separator="true">,</mo><mo>…</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B^c=\{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, \ldots\} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">10</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">11</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">13</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">14</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">}</span></span></span></span></span></p> <ol start="3"> <li>The union of the sets <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the set of all elements that are in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> or in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</li> </ol> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>7</mn><mo separator="true">,</mo><mn>8</mn><mo separator="true">,</mo><mn>9</mn><mo separator="true">,</mo><mn>10</mn><mo separator="true">,</mo><mo>…</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A \cup B=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \ldots\} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">9</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">10</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">}</span></span></span></span></span></p> <p>The complement of the union of the sets <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the set of all elements that are not in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> or in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><msup><mo stretchy="false">)</mo><mi>c</mi></msup><mo>=</mo><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">(A \cup B)^c=\emptyset </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span></span></p> <ol start="4"> <li>The intersection of the sets <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the set of all elements that are in both <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</li> </ol> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mn>3</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>9</mn><mo separator="true">,</mo><mn>12</mn><mo separator="true">,</mo><mo>…</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A \cap B=\{3, 6, 9, 12, \ldots\} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">9</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">12</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">}</span></span></span></span></span></p> <p>The complement of the intersection of the sets <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> is the set of all elements that are not in both <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><msup><mo stretchy="false">)</mo><mi>c</mi></msup><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo separator="true">,</mo><mn>7</mn><mo separator="true">,</mo><mn>8</mn><mo separator="true">,</mo><mn>10</mn><mo separator="true">,</mo><mn>11</mn><mo separator="true">,</mo><mn>13</mn><mo separator="true">,</mo><mn>14</mn><mo separator="true">,</mo><mo>…</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(A \cap B)^c=\{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, \ldots\} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">10</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">11</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">13</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">14</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">}</span></span></span></span></span></p>$$