Determine whether
~ [ ~(p v ~ q) <-> p v ~ q. Explain the method(s) you used to determine your answer
i would use a truth table
To determine whether ~ [ ~(p v ~ q) <-> p v ~ q is true or false, you can indeed use a truth table. Here's a step-by-step explanation of how to construct and analyze the truth table for this statement.
Step 1: Identify the atomic propositions
In this statement, we have two atomic propositions: p and q. These are the basic propositions that can either be true or false.
Step 2: Create the truth table columns
Create columns for p, q, ~(p v ~q), p v ~q, ~(p v ~q) <-> p v ~q, and ~[~(p v ~q) <-> p v ~q].
Step 3: Fill in the truth values for p and q
Since p and q can each have two truth values (true or false), you will have four rows in your truth table.
Step 4: Calculate the truth values for ~(p v ~q) and p v ~q
For each row, apply the logical operators ~(negation), v (disjunction), and the parentheses to calculate the truth values for ~(p v ~q) and p v ~q.
Step 5: Determine the truth value for ~(p v ~q) <-> p v ~q
Apply the logical operator <-> (biconditional) to compare the truth values of ~(p v ~q) and p v ~q for each row.
Step 6: Determine the truth value for ~[~(p v ~q) <-> p v ~q]
Finally, apply the negation operator ~ to the truth values obtained from the previous step to determine the overall truth value for the entire statement ~ [ ~(p v ~ q) <-> p v ~ q].
Step 7: Analyze the truth table
Examine the truth values in the last column of the truth table. If all values in the column are true, the statement is true. If there is at least one false value, the statement is false.
By following these steps and completing the truth table, you can determine whether the given statement is true or false.