Question

Let \(\mathrm{P}\) and \(\mathrm{Q}\) be two statements, then \(\sim(\sim \mathrm{P} \wedge \mathrm{Q}) \wedge(\mathrm{P} \vee \mathrm{Q})\) is logically equivalent to (1) \(Q\) (2) \(\mathrm{P}\) (3) \(P \vee Q\) (4) \(\mathrm{P} \wedge \mathrm{Q}\)

Short answer

The correct answer is (2) \text{P} \.

Step-by-step solution

- Understand the Problem

Given the complex logical statement \((eg(eg \text{P} \wedge \text{Q}) \wedge (\text{P} \vee \text{Q}))\), determine the logically equivalent statement from the options provided.

- Simplify the Inner Negation

The innermost part is \((eg \text{P} \wedge \text{Q})\). Apply De Morgan's law to negate the inner statement: \(eg(eg \text{P} \wedge \text{Q}) \equiv (eg(eg \text{P}) \vee eg \text{Q}) \equiv \text{P} \vee eg \text{Q}\).

- Substitute Back and Further Simplify

Replace \(eg(eg \text{P} \wedge \text{Q})\) with \( \text{P} \vee eg \text{Q} \). The statement now becomes \( (\text{P} \vee eg \text{Q}) \wedge (\text{P} \vee \text{Q}) \).

- Distribute and Analyze

Next, use the distributive property of logical conjunction over disjunction: \( (\text{P} \vee eg \text{Q}) \wedge (\text{P} \vee \text{Q}) \equiv (\text{P} \vee (eg \text{Q} \wedge \text{Q})) \). Since \( (eg \text{Q} \wedge \text{Q}) \) is always false, this simplifies to \( (\text{P} \vee \text{false}) \equiv \text{P} \).

- Conclude the Equivalent Statement

The simplified equivalent statement is logically equivalent to \( \text{P} \). Therefore, the correct answer is (2) \text{P} \.

Key concepts

De Morgan's laws

De Morgan's laws are essential rules in Boolean algebra used for negating logical expressions. These laws help in simplifying complex logical statements, especially when dealing with negations.
De Morgan's laws state that:

1. The negation of a conjunction is the disjunction of the negations:
\[ eg (P \text{ and } Q) \text{ is equivalent to } (eg P \text{ or } eg Q). \]
2. The negation of a disjunction is the conjunction of the negations:
\[ eg (P \text{ or } Q) \text{ is equivalent to } (eg P \text{ and } eg Q). \]

In the given problem, we used the first law to transform the expression \( eg (eg P \text{ and } Q) \) into \( P \text{ or } eg Q \). This simplification makes it easier to handle and analyze the logical statement. Understanding these laws is crucial for students aiming to master Boolean algebra.

Logical conjunction

Logical conjunction is an operation in Boolean algebra that corresponds to the word 'and' in everyday language. It combines two statements and returns true only if both statements are true.
An example of conjunction is: \( P \text{ and } Q \), represented as \( P \text{ \textbackslash{}wedge } Q \). Here are some facts about conjunction:

• \( P \text{ \textbackslash{}wedge } Q \) is true if and only if both \( P \) and \( Q \) are true.
• If either \( P \) or \( Q \) is false, the entire conjunction is false.
• In the problem, conjunction is used within the expression \( (eg P \text{ \textbackslash{}wedge } Q) \).

Understanding conjunction helps us see why \( (eg Q \text{ \textbackslash{}wedge } Q) \) simplifies to false, aiding further simplification of the entire logical statement.

Logical disjunction

Logical disjunction is an operation in Boolean algebra corresponding to the word 'or'. It combines two statements and returns true if at least one of the statements is true.
Disjunction is represented as \( P \text{ \textbackslash{}vee } Q \) where \( P \) and \( Q \) are individual statements. Some key points about disjunction:

• \( P \text{ \textbackslash{}vee } Q \) is true if either \( P \) or \( Q \) is true.
• It only returns false if both statements \( P \) and \( Q \) are false.
• In our problem, we see disjunction in expressions like \( P \text{ \textbackslash{}vee } Q \) and \( P \text{ \textbackslash{}vee } eg Q \).

Using disjunction helps us understand how the simplified statement \( P \text{ \textbackslash{}vee } eg Q \) can be further combined with another disjunction to reach the final equivalent expression.