Truth Table for the Compound Statement

Oct 13, 2025 by windailys.com 81 views

Introduction

In propositional logic, a truth table is a mathematical table used to determine the truth value of a compound statement for every possible combination of truth values of its component statements. The given compound statement is $(p \Rightarrow \sim q) \wedge (p \vee q)$ . In this article, we will complete the truth table for the given compound statement and then decide which statements within the given statement are logically equivalent.

Truth Table for the Compound Statement

To complete the truth table for the compound statement, we need to consider all possible combinations of truth values of the component statements $p$ and $q$ . The truth table is as follows:

$p$	$q$	$\sim q$	$p \Rightarrow \sim q$	$p \vee q$	$(p \Rightarrow \sim q) \wedge (p \vee q)$
T	T	F	F	T	F
T	F	T	T	T	T
F	T	F	T	T	T
F	F	T	T	F	F

Explanation of the Truth Table

In the truth table, we have considered all possible combinations of truth values of the component statements $p$ and $q$ . The truth values of the compound statements are determined as follows:

$\sim q$ is the negation of $q$ , which is true when $q$ is false and false when $q$ is true.
$p \Rightarrow \sim q$ is the implication of $p$ by $\sim q$ , which is true when $p$ is false or $\sim q$ is true. In other words, it is true when $p$ is false or $q$ is false.
$p \vee q$ is the disjunction of $p$ and $q$ , which is true when either $p$ or $q$ is true.
$(p \Rightarrow \sim q) \wedge (p \vee q)$ is the conjunction of $p \Rightarrow \sim q$ and $p \vee q$ , which is true when both $p \Rightarrow \sim q$ and $p \vee q$ are true.

Logical Equivalence

Two statements are logically equivalent if they have the same truth value for every possible combination of truth values of their component statements. In other words, two statements are logically equivalent if their truth tables are identical.

From the truth table, we can see that the compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ is true when $p$ is true and $q$ is false, and when $p$ is false and $q$ is true. This means that the compound statement is true when $p$ and $q$ have different truth values.

Conclusion

In conclusion, we have completed the truth table for the given compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ and determined which statements within the given statement are logically equivalent. The compound statement is true when $p$ and $q$ have different truth values. This means that the compound statement is logically equivalent to the statement $(p \wedge \sim q) \vee (q \wedge \sim p)$ .

Final Answer

The final answer is that the compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ is logically equivalent to the statement $(p \wedge \sim q) \vee (q \wedge \sim p)$ .

References

[1] "Propositional Logic" by Stanford Encyclopedia of Philosophy
[2] "Truth Tables" by Math Is Fun
[3] "Logical Equivalence" by Khan Academy

Introduction

In our previous article, we completed the truth table for the given compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ and determined which statements within the given statement are logically equivalent. In this article, we will answer some frequently asked questions related to the topic.

Q1: What is a truth table in propositional logic?

A1: A truth table is a mathematical table used to determine the truth value of a compound statement for every possible combination of truth values of its component statements.

Q2: How do you complete a truth table for a compound statement?

A2: To complete a truth table for a compound statement, you need to consider all possible combinations of truth values of the component statements. You then determine the truth value of the compound statement for each combination of truth values.

Q3: What is the difference between a conjunction and a disjunction in propositional logic?

A3: A conjunction is a compound statement that is true when both component statements are true. A disjunction is a compound statement that is true when at least one of the component statements is true.

Q4: How do you determine which statements are logically equivalent?

A4: Two statements are logically equivalent if they have the same truth value for every possible combination of truth values of their component statements. In other words, two statements are logically equivalent if their truth tables are identical.

Q5: What is the significance of logical equivalence in propositional logic?

A5: Logical equivalence is significant in propositional logic because it allows us to simplify complex statements by replacing them with equivalent statements that are easier to work with.

Q6: Can you give an example of a compound statement that is logically equivalent to the given compound statement?

A6: Yes, the compound statement $(p \wedge \sim q) \vee (q \wedge \sim p)$ is logically equivalent to the given compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ .

Q7: How do you use truth tables to determine the validity of an argument?

A7: To use truth tables to determine the validity of an argument, you need to construct a truth table for the argument and then determine whether the conclusion is true for every possible combination of truth values of the premises.

Q8: What are some common mistakes to avoid when completing a truth table?

A8: Some common mistakes to avoid when completing a truth table include:

Failing to consider all possible combinations of truth values of the component statements
Making errors in determining the truth value of the compound statement for each combination of truth values
Failing to check for logical equivalence between statements

Q9: Can you give an example of a compound statement that is not logically equivalent to the given compound statement?

A9: Yes, the compound statement $(p \wedge q) \vee (p \vee \sim q)$ is not logically equivalent to the given compound statement $(p \Rightarrow \sim q) \wedge (p \vee q)$ .

Q10: How do you use truth tables to determine the validity of a logical argument in a real-world context?

A10: To use truth tables to determine the validity of a logical argument in a real-world context, you need to identify the premises and conclusion of the argument, construct a truth table for the argument, and then determine whether the conclusion is true for every possible combination of truth values of the premises.

Conclusion

In conclusion, we have answered some frequently asked questions related to completing the truth table for the given compound statement and deciding which statements are logically equivalent. We hope that this article has provided a helpful resource for those who are studying propositional logic.