2.) Construct a truth table for (p^~q)-->q.
To construct a truth table for the logical expression (p^~q) --> q, follow these steps:
1. Determine the number of inputs: The given expression contains two variables, p and q. Since each variable can take on two possible truth values (true or false), there will be 2^2 = 4 rows in our truth table.
2. Create columns for variables: Create two columns for p and q.
p | q |
--|---|
? | ? |
3. Calculate the truth values of ~q: The tilde (~) represents the negation or NOT operator. In this step, negate the values of q to obtain ~q.
p | q | ~q |
--|---|----|
? | ? | ? |
Fill in the ~q column according to the following table:
q | ~q |
--|----|
T | F |
F | T |
4. Calculate the truth values of (p^~q): The carrot (^) symbol represents the logical AND operator. In this step, calculate the conjunction of p and ~q.
p | q | ~q | p^~q |
--|---|----|------|
? | ? | ? | ? |
Fill in the p^~q column by evaluating the conjunction for each row:
p | q | ~q | p^~q |
--|---|----|------|
T | T | F | F |
T | F | T | T |
F | T | F | F |
F | F | T | F |
5. Determine the values of the entire expression: The implication operator (--> or ⇒) represents the logical implication. In this step, evaluate the implication (p^~q) --> q.
p | q | ~q | p^~q | (p^~q) --> q |
--|---|----|------|-------------|
T | T | F | F | T |
T | F | T | T | F |
F | T | F | F | T |
F | F | T | F | T |
The completed truth table shows the resulting values for each row of the expression (p^~q) --> q.