use truth table to verify:
1.Distributive law
P conjunction ( q disjunction r )= ( p conjunction q ) disjunction ( p conjunction r)
2.Associative law
( p disjunction q) disjunction r = p disjunction ( p disjunction r )
( p conjunction q) conjunction r= p conjunction ( q conjunction r)
To verify both the distributive and associative laws using a truth table, we need to create a truth table for each expression and compare the resulting truth values for all possible combinations of truth and false values for the variables involved.
1. Distributive Law:
To verify the distributive law, we'll compare the truth values for the expressions: P ∧ (Q ∨ R) and (P ∧ Q) ∨ (P ∧ R).
Start by creating a truth table with columns for P, Q, R, P ∧ (Q ∨ R), (P ∧ Q) ∨ (P ∧ R). Assign truth (T) and false (F) values to the variables P, Q, and R in all possible combinations.
Here is the truth table:
| P | Q | R | Q ∨ R | P ∧ (Q ∨ R) | (P ∧ Q) ∨ (P ∧ R) |
|---|---|---|-------|-------------|------------------|
| T | T | T | T | T | T |
| T | T | F | T | T | T |
| T | F | T | T | T | T |
| T | F | F | F | F | F |
| F | T | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | T | F | F |
| F | F | F | F | F | F |
Comparing the truth values for the last two columns, we can see that they are the same for all the combinations. Therefore, we can conclude that the statement P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R) is true.
2. Associative Law:
To verify the associative laws, we'll compare the truth values for the expressions: (P ∨ Q) ∨ R and P ∨ (Q ∨ R) for the first case, and (P ∧ Q) ∧ R and P ∧ (Q ∧ R) for the second case.
Create a truth table with columns for P, Q, R, (P ∨ Q) ∨ R, P ∨ (Q ∨ R), (P ∧ Q) ∧ R, and P ∧ (Q ∧ R). Assign truth (T) and false (F) values to the variables P, Q, and R in all possible combinations.
Here is the truth table:
| P | Q | R | P ∨ Q | (P ∨ Q) ∨ R | P ∨ (Q ∨ R) | (P ∧ Q) ∧ R | P ∧ (Q ∧ R) |
|---|---|---|-------|-------------|-------------|-------------|-------------|
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | F |
| T | F | T | T | T | T | T | F |
| T | F | F | T | T | T | F | F |
| F | T | T | T | T | T | F | F |
| F | T | F | T | F | F | F | F |
| F | F | T | F | T | F | F | F |
| F | F | F | F | F | F | F | F |
Comparing the truth values for the last four columns, we can see that they are the same for all the combinations. Therefore, we can conclude that the statements (P ∨ Q) ∨ R = P ∨ (Q ∨ R) and (P ∧ Q) ∧ R = P ∧ (Q ∧ R) are true.