To construct the truth table for the compound statement $\sim p \lor q$, let's calculate each column step by step.

Truth Table Layout

The table has the columns:

1. $p$ (truth value of the first variable)
2. $q$ (truth value of the second variable)
3. $\sim p$ (negation of $p$)
4. $\sim p \lor q$ (logical OR between $\sim p$ and $q$)

Truth Table Construction

$p$	$q$	$\sim p$	$\sim p \lor q$
T	T	F	T
T	F	F	F
F	T	T	T
F	F	T	T

Step-by-Step Explanation

1. Column $p$ and $q$: The truth values of $p$ and $q$ are taken as all possible combinations of $T$ and $F$ (true and false). There are four combinations: $(T, T), (T, F), (F, T), (F, F)$.
2. Column $\sim p$: Negate the values of $p$. If $p = T$, then $\sim p = F$; if $p = F$, then $\sim p = T$.
3. Column $\sim p \lor q$: Apply the logical OR ($\lor$) operation between $\sim p$ and $q$. Logical OR is true if at least one operand is true:
   • $F \lor T = T$
   • $F \lor F = F$
   • $T \lor T = T$
   • $T \lor F = T$

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Do you want a more detailed explanation of how logical OR works or any part of the table? Here are related questions:

1. What does the symbol $\lor$ represent in logic?
2. How is the negation $\sim p$ calculated?
3. What are the basic truth table rules for $\lor$ (OR)?
4. Can you provide an example with different variables to practice this?
5. How does this relate to circuits in digital logic?

Tip: For compound logical statements, build the table incrementally to avoid errors!