Construct a truth table for ~q^p
To construct a truth table for the logical expression ~q^p, we need to consider the possible combinations of truth values for the variables q and p.
Since there are two variables, q and p, each variable can take either the value true (T) or false (F). Therefore, there are four possible combinations of truth values for the variables:
1. q = T, p = T
2. q = T, p = F
3. q = F, p = T
4. q = F, p = F
For each combination, we will evaluate the logical expression ~q^p using the following steps:
1. Determine the value of ~q: The negation (~) of q is simply the opposite of its truth value.
2. Evaluate the conjunction (^): The logical AND (^) operation returns true (T) only if both operands are true (T).
Let's construct the truth table step by step:
```
| q | p | ~q | ~q^p |
| --- | --- | --- | ---- |
| T | T | F | F |
| T | F | F | F |
| F | T | T | T |
| F | F | T | F |
```
In the truth table, we first list all the possible combinations of truth values for q and p. Then, we determine the value of ~q for each combination, followed by evaluating the expression ~q^p. The final column represents the result of the logical expression.
Therefore, the truth table for ~q^p is as shown above.