Fill In The Missing Values In The Table. Fill In The Blanks Below. \[ \begin{tabular}{|c|c|c|c|c|c|} \hline P$ & Q Q Q & R R R & ∼ P \sim P ∼ P & ∼ Q \sim Q ∼ Q & ( ∼ P ∨ ∼ Q ) → R (\sim P \vee \sim Q) \rightarrow R ( ∼ P ∨ ∼ Q ) → R \ \hline T T T & T T T & T T T & F F F & F F F &

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Introduction

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In this article, we will be filling in the missing values in a table related to mathematical logic. The table contains various logical statements and their corresponding truth values. We will use the rules of logical operations to determine the missing values.

The Table

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$T$	$T$	$T$	$F$	$F$
$T$	$F$	$T$	$F$	$T$
$F$	$T$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$T$
$T$	$T$	$F$	$F$	$F$
$T$	$F$	$F$	$F$	$T$
$F$	$T$	$F$	$T$	$F$
$F$	$F$	$F$	$T$	$T$

Filling in the Missing Values

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To fill in the missing values, we need to understand the rules of logical operations. The table contains the following logical statements:

• $\sim p$: The negation of $p$, which is true if $p$ is false and false if $p$ is true.
• $\sim q$: The negation of $q$, which is true if $q$ is false and false if $q$ is true.
• $(\sim p \vee \sim q) \rightarrow r$: The implication of the disjunction of the negations of $p$ and $q$ to $r$. This statement is true if the disjunction of the negations of $p$ and $q$ is true, and $r$ is true. Otherwise, it is false.

Using these rules, we can fill in the missing values in the table.

Row 1

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$T$	$T$	$T$	$F$	$F$	$T$

Since $p$ and $q$ are both true, $\sim p$ and $\sim q$ are both false. Therefore, the disjunction of the negations of $p$ and $q$ is false. Since $r$ is true, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is true.

Row 2

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$T$	$F$	$T$	$F$	$T$	$T$

Since $p$ is true and $q$ is false, $\sim p$ is false and $\sim q$ is true. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is true, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is true.

Row 3

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$F$	$T$	$T$	$T$	$F$	$T$

Since $p$ is false and $q$ is true, $\sim p$ is true and $\sim q$ is false. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is true, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is true.

Row 4

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$F$	$F$	$T$	$T$	$T$	$T$

Since $p$ is false and $q$ is false, $\sim p$ is true and $\sim q$ is true. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is true, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is true.

Row 5

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$T$	$T$	$F$	$F$	$F$	$F$

Since $p$ and $q$ are both true, $\sim p$ and $\sim q$ are both false. Therefore, the disjunction of the negations of $p$ and $q$ is false. Since $r$ is false, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is false.

Row 6

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$T$	$F$	$F$	$F$	$T$	$F$

Since $p$ is true and $q$ is false, $\sim p$ is false and $\sim q$ is true. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is false, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is false.

Row 7

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$F$	$T$	$F$	$T$	$F$	$F$

Since $p$ is false and $q$ is true, $\sim p$ is true and $\sim q$ is false. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is false, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is false.

Row 8

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$p$	$q$	$r$	$\sim p$	$\sim q$	$(\sim p \vee \sim q) \rightarrow r$
$F$	$F$	$F$	$T$	$T$	$F$

Since $p$ is false and $q$ is false, $\sim p$ is true and $\sim q$ is true. Therefore, the disjunction of the negations of $p$ and $q$ is true. Since $r$ is false, the implication of the disjunction of the negations of $p$ and $q$ to $r$ is false.

Conclusion

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In this article, we filled in the missing values in a table related to mathematical logic. We used the rules of logical operations to determine the missing values. The table contains various logical statements and their corresponding truth values. We hope that this article has provided a clear understanding of the rules of logical operations and how to apply them to fill in missing values in a table.

References

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• [1] "Logical Operations" by Wikipedia
• [2] "Mathematical Logic" by Stanford Encyclopedia of Philosophy

Discussion

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The table in this article is a simple example of how to apply the rules of logical operations to fill in missing values. In real-world applications, the tables may be more complex and contain more variables. However, the principles of logical operations remain the same.

The rules of logical operations are essential in mathematics and computer science. They are used to determine the truth values of logical statements and to make decisions based on those statements. In this article, we have seen how to apply the rules of logical operations to fill in missing values in a table.

We hope that this article has provided a clear understanding of the rules of logical operations and how to apply them to fill in missing values in a table. If you have any questions or

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Introduction

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In our previous article, we filled in the missing values in a table related to mathematical logic. We used the rules of logical operations to determine the missing values. In this article, we will answer some frequently asked questions (FAQs) related to the table and the rules of logical operations.

Q&A

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Q: What is the difference between $\sim p$ and $p$?

A: $\sim p$ is the negation of $p$, which is true if $p$ is false and false if $p$ is true. $p$ is the original statement, which can be true or false.

Q: How do you determine the truth value of $(\sim p \vee \sim q) \rightarrow r$?

A: To determine the truth value of $(\sim p \vee \sim q) \rightarrow r$, you need to follow these steps:

1. Determine the truth values of $\sim p$ and $\sim q$.
2. Determine the truth value of the disjunction of $\sim p$ and $\sim q$.
3. Determine the truth value of the implication of the disjunction of $\sim p$ and $\sim q$ to $r$.

Q: What is the difference between $\sim p \vee \sim q$ and $\sim (p \wedge q)$?

A: $\sim p \vee \sim q$ is the disjunction of the negations of $p$ and $q$. $\sim (p \wedge q)$ is the negation of the conjunction of $p$ and $q$. These two statements are not equivalent.

Q: How do you determine the truth value of $\sim (p \wedge q)$?

A: To determine the truth value of $\sim (p \wedge q)$, you need to follow these steps:

1. Determine the truth values of $p$ and $q$.
2. Determine the truth value of the conjunction of $p$ and $q$.
3. Determine the truth value of the negation of the conjunction of $p$ and $q$.

Q: What is the difference between $p \rightarrow q$ and $\sim p \vee q$?

A: $p \rightarrow q$ is the implication of $p$ to $q$. $\sim p \vee q$ is the disjunction of the negation of $p$ and $q$. These two statements are not equivalent.

Q: How do you determine the truth value of $p \rightarrow q$?

A: To determine the truth value of $p \rightarrow q$, you need to follow these steps:

1. Determine the truth values of $p$ and $q$.
2. Determine the truth value of the implication of $p$ to $q$.

Q: What is the difference between $\sim p \wedge \sim q$ and $\sim (p \vee q)$?

A: $\sim p \wedge \sim q$ is the conjunction of the negations of $p$ and $q$. $\sim (p \vee q)$ is the negation of the disjunction of $p$ and $q$. These two statements are not equivalent.

Q: How do you determine the truth value of $\sim (p \vee q)$?

A: To determine the truth value of $\sim (p \vee q)$, you need to follow these steps:

1. Determine the truth values of $p$ and $q$.
2. Determine the truth value of the disjunction of $p$ and $q$.
3. Determine the truth value of the negation of the disjunction of $p$ and $q$.

Conclusion

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In this article, we have answered some frequently asked questions (FAQs) related to the table and the rules of logical operations. We hope that this article has provided a clear understanding of the rules of logical operations and how to apply them to fill in missing values in a table.

References

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• [1] "Logical Operations" by Wikipedia
• [2] "Mathematical Logic" by Stanford Encyclopedia of Philosophy

Discussion

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The rules of logical operations are essential in mathematics and computer science. They are used to determine the truth values of logical statements and to make decisions based on those statements. In this article, we have seen how to apply the rules of logical operations to fill in missing values in a table and how to answer some frequently asked questions (FAQs) related to the table and the rules of logical operations.

We hope that this article has provided a clear understanding of the rules of logical operations and how to apply them to fill in missing values in a table. If you have any questions or need further clarification, please don't hesitate to contact us.