To create a truth table for the expression $\sim q \to (p \vee \sim r)$, we first rewrite the implication $A \to B$ using its equivalent form: $\sim A \vee B$. Thus, the given expression can be rewritten as:

$\sim q \to (p \vee \sim r) \equiv \sim (\sim q) \vee (p \vee \sim r) \equiv q \vee (p \vee \sim r).$

This means we need to evaluate the expression $q \vee (p \vee \sim r)$. Let's construct the truth table step by step:

Variables

• $p$
• $q$
• $r$

Derived Columns

1. $\sim q$
2. $\sim r$
3. $p \vee \sim r$
4. $\sim q \to (p \vee \sim r)$

Truth Table

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

Explanation

1. Column $\sim q$: The negation of $q$.
2. Column $\sim r$: The negation of $r$.
3. Column $p \vee \sim r$: The logical OR between $p$ and $\sim r$.
4. Column $\sim q \to (p \vee \sim r)$: Evaluated as $\sim q \vee (p \vee \sim r)$, combining the results.

Let me know if you'd like more detailed steps for each column! 😊

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5 Related Questions

1. What is the difference between logical implication ($\to$) and equivalence ($\leftrightarrow$)?
2. How do truth tables help verify logical equivalences?
3. Can we derive $p \vee q$ from $\sim q \to p$? Why or why not?
4. What is the purpose of using De Morgan's Laws in logical expressions?
5. How does $\sim (\sim p)$ simplify in logical expressions?

Tip

When constructing a truth table, ensure each variable has $2^n$ rows, where $n$ is the number of distinct variables. This ensures all combinations are covered!