First, let’s define the variables used in the statements:
- p, q, and r are propositional variables representing statements that can be either true or false.
Now, let’s construct the truth tables for each statement:
(i) – (pɅq)Ʌ (-r)
To construct the truth table for this statement, we need to consider all possible combinations of truth values for p, q, and r. There are 2^3 = 8 possible combinations, which we can list as follows:
| p | q | r | -r | pɅq | (pɅq)Ʌ(-r) |
|---|---|---|---|---|---|
| T | T | T | F | T | F |
| T | T | F | T | T | T |
| T | F | T | F | F | F |
| T | F | F | T | F | F |
| F | T | T | F | F | F |
| F | T | F | T | F | F |
| F | F | T | F | F | F |
| F | F | F | T | F | F |
In the table above, the column labeled “-r” represents the negation of r, and the column labeled “pɅq” represents the conjunction (logical AND) of p and q. The column labeled “(pɅq)Ʌ(-r)” represents the conjunction of p and q, negated, and then conjuncted with the negation of r.
(ii) – (pɅ-q) v (r)
To construct the truth table for this statement, we need to consider all possible combinations of truth values for p, q, and r. There are 2^3 = 8 possible combinations, which we can list as follows:
| p | q | r | -q | pɅ-q | -pɅq | (pɅ-q)v(r) |
|---|---|---|---|---|---|---|
| T | T | T | F | T | F | T |
| T | T | F | F | F | F | F |
| T | F | T | T | F | F | T |
| T | F | F | T | F | F | F |
| F | T | T | F | F | F | T |
| F | T | F | F | F | F | F |
| F | F | T | T | F | F | T |
| F | F | F | T | F | F | F |
In the table above, the column labeled “-q” represents the negation of q, and the column labeled “pɅ-q” represents the conjunction of p and the negation of q. The column labeled “-pɅq” represents the negation of the conjunction of p and q. Finally, the column labeled “(pɅ-q)v(r)” represents the disjunction (logical OR) of the conjunction of p and the negation of q, and r.