Let's break down and analyze both of these logical expressions step by step.

Expression 1: $(p \lor q) \rightarrow r$

This is a conditional statement where:

• $p$ and $q$ are propositions.
• $\lor$ represents the logical OR.
• $\rightarrow$ represents implication.

Truth Table for $(p \lor q) \rightarrow r$:

$p$	$q$	$p \lor q$	$r$	$(p \lor q) \rightarrow r$
T	T	T	T	T
T	T	T	F	F
T	F	T	T	T
T	F	T	F	F
F	T	T	T	T
F	T	T	F	F
F	F	F	T	T
F	F	F	F	T

• $(p \lor q)$ becomes true if either $p$ or $q$ is true, and the implication $(p \lor q) \rightarrow r$ is only false when $(p \lor q)$ is true and $r$ is false.

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Expression 2: $\neg (q \land r) \leftrightarrow (p \land q)$

This is a biconditional statement where:

• $\neg$ represents the negation (NOT).
• $\land$ represents logical AND.
• $\leftrightarrow$ represents equivalence (if and only if).

Truth Table for $\neg (q \land r) \leftrightarrow (p \land q)$:

$p$	$q$	$r$	$q \land r$	$\neg (q \land r)$	$p \land q$	$\neg (q \land r) \leftrightarrow (p \land q)$
T	T	T	T	F	T	F
T	T	F	F	T	T	T
T	F	T	F	T	F	F
T	F	F	F	T	F	F
F	T	T	T	F	F	T
F	T	F	F	T	F	F
F	F	T	F	T	F	T
F	F	F	F	T	F	T

The biconditional $\leftrightarrow$ is true when both sides of the statement are either both true or both false.

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Summary:

1. $(p \lor q) \rightarrow r$ is a conditional statement that is only false when $p \lor q$ is true and $r$ is false.
2. $\neg (q \land r) \leftrightarrow (p \land q)$ is a biconditional statement that compares the truth values of $\neg (q \land r)$ and $(p \land q)$, and it is true when both sides have the same truth value.

Would you like to dive deeper into any part of this, or see some specific examples?

Follow-up Questions:

1. What is the meaning of a biconditional statement?
2. How can the implication $\rightarrow$ be used in logical proofs?
3. What are De Morgan's laws in logic?
4. How does the truth table help in understanding logical equivalences?
5. Can you give an example where these logical statements apply in real-world scenarios?

Tip: The biconditional operator $\leftrightarrow$ can be thought of as "both sides must either be true or both false."