Construct a truth table for
(p ^ ~q) <-->q
^ is meant to be a conjunction symbol
p......q......~q......p ^ ~q......(p ^ ~q) <--> q
T......T.......F..........F.....................F
T......F.......T..........T.....................F
F......T.......F..........F.....................F
F......F.......T..........F.....................T
To construct a truth table for the given expression (p ^ ~q) <--> q, we need to evaluate the expression for all possible combinations of truth values for p and q.
First, let's focus on the sub-expression (p ^ ~q). Here's how we can evaluate this expression:
1. Create two columns, one for p and one for q.
2. Fill in all possible combinations of truth values for p and q in the columns.
3. In the q column, write the negation of the current q value.
4. Apply the conjunction operation (^) between the corresponding p and ~q values, and write the result in a new column (let's call it R1).
Next, let's evaluate the sub-expression (p ^ ~q) <--> q:
5. Create another column (let's call it R2).
6. Apply the biconditional operation (<-->) between the values in the R1 column and the q column, and write the result in the R2 column.
Finally, the truth table will consist of four columns: p, q, R1, and R2.
Here's how the complete truth table looks:
| p | q | ~q | (p ^ ~q) | [(p ^ ~q) <--> q] |
| --- | --- | ---- | ---------- | ------------------ |
| T | T | F | F | F |
| T | F | T | T | F |
| F | T | F | F | F |
| F | F | T | F | F |
In the truth table above, T represents true, and F represents false.