4. Construct a truth table for
q <-->(p^ ~q). Be sure to include all intermediate steps in your table.
To construct a truth table for the logical expression q <-> (p ^ ~q), we need to consider all possible combinations of truth values for the variables p and q and evaluate the expression for each combination.
The expression q <-> (p ^ ~q) involves three logical operators: <-> (biconditional), ^ (conjunction), and ~ (negation). Let's break down the expression step by step:
1. Start by listing all possible combinations of truth values for p and q.
- There are two possibilities for each variable: p can be either true (T) or false (F), and q can be either true (T) or false (F).
- So, we have a total of 2 * 2 = 4 possible combinations:
- Combination 1: p = T, q = T
- Combination 2: p = T, q = F
- Combination 3: p = F, q = T
- Combination 4: p = F, q = F
2. Evaluate the expression q <-> (p ^ ~q) for each combination.
- For combination 1 (p = T, q = T):
- ~q = ~T = F
- (p ^ ~q) = (T ^ F) = F
- q <-> (p ^ ~q) = T <-> F = F
- For combination 2 (p = T, q = F):
- ~q = ~F = T
- (p ^ ~q) = (T ^ T) = T
- q <-> (p ^ ~q) = F <-> T = F
- For combination 3 (p = F, q = T):
- ~q = ~T = F
- (p ^ ~q) = (F ^ F) = F
- q <-> (p ^ ~q) = T <-> F = F
- For combination 4 (p = F, q = F):
- ~q = ~F = T
- (p ^ ~q) = (F ^ T) = F
- q <-> (p ^ ~q) = F <-> F = T
3. Organize the results in a truth table.
- The truth table will have four rows, one for each combination, and two columns, one for the variable q and one for the expression q <-> (p ^ ~q).
- The truth table will look like this:
| p | q | (p ^ ~q) | q <-> (p ^ ~q) |
--------------------------------------
| T | T | F | F |
| T | F | T | F |
| F | T | F | F |
| F | F | F | T |
In summary, the truth table for the logical expression q <-> (p ^ ~q) is as shown above.