how would i construct a truth table for (p and ~q) double arrow q (sorry inexperienced with the symbols and keyboard shortcuts).

To construct a truth table for the logical expression "(p and ~q) double arrow q," follow these steps:

1. Identify all the distinct variables in the expression. In this case, we have two variables: p and q.

2. Determine the total number of rows needed for the truth table. Since we have two variables, there will be 2^2 = 4 rows in the truth table.

3. Create a column for each variable, including its possible truth values. In this case, we will have two columns, one for p and one for q. Each column will have two rows, representing the possible truth values: true (T) and false (F).

4. Create a column for the expression "(p and ~q)". This column will represent the conjunction of p and the negation of q. Use the truth values from the previous columns to determine the truth values of this expression.

To do this, apply the logical AND operator (^) to p and the negation of q. If p is true and q is false, ~q will be true, resulting in the expression (p and ~q) also being true. If p is false, regardless of the truth value of q, the entire expression will be false.

5. Create a column for the expression "(p and ~q) double arrow q". This column will represent the double implication between the expression "(p and ~q)" and q. Again, use the truth values from the previous columns to determine the truth values of this expression.

To evaluate the double implication, compare the truth values of the two expressions: "(p and ~q)" and q. If they have the same truth value (both true or both false), then the double implication is true. If they have different truth values, the double implication is false.

So, in our case, compare the truth values of the "(p and ~q)" column and the q column. If they are the same, write T in the "(p and ~q) double arrow q" column; if they are different, write F.

6. Complete the truth table by filling in the remaining columns with the truth values you have calculated.

Here is the truth table for the logical expression "(p and ~q) double arrow q":

p | q | (p and ~q) | (p and ~q) double arrow q
----------------------------------------------
T | T | F | T
T | F | F | F
F | T | F | F
F | F | F | T

You can see that the truth values for each row were determined by evaluating the expressions according to the defined logical operations.