The completed truth table for the two compound propositions ~ (p^q) and ~p V~q is as follows:
p q ~ (p^q) ~p V~q
T T F T
T F T T
F T T T
T T T T .
Partially completed truth table for two propositional logic operators: logical negation (~) and logical disjunction (V).
The truth table shows the truth values of the compound propositions ~ (p^q) and ~p V~q for all possible combinations of truth values for the atomic propositions p and q.
Logical negation is an operator that takes a single proposition as input and returns a new proposition that is true if the input proposition is false, and false if the input proposition is true.
The truth table for logical negation is as follows:
p ~p
T F
F T
Logical disjunction is an operator that takes two propositions as input and returns a new proposition that is true if either or both of the input propositions are true, and false only if both input propositions are false.
The truth table for logical disjunction is as follows:
p q p V q
T T T
T F T
F T T
F F F
To complete the truth table for the compound proposition ~p V~q, we simply need to apply the truth table for logical disjunction to the negated atomic propositions ~p and ~q.
The truth table for ~p V~q is as follows:
p q ~p ~q ~p V~q
T T F F T
T F F T T
F T T F T
F F T T T