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 the truth of the given statements using truth tables, we need to create a truth table for each expression and compare the resulting truth values for all possible combinations of truth values of the propositional variables (P, Q, and R).
1. Distributive Law:
Statement: P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)
Create a truth table for both sides of the equation:
| P | Q | R | Q ∨ R | P ∧ (Q ∨ R) | P ∧ Q | P ∧ R | (P ∧ Q) ∨ (P ∧ R) |
|---|---|---|-------|------------|-------|-------|------------------|
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | T |
| T | F | T | T | T | F | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | F | F | F | F |
| F | T | F | T | F | F | F | F |
| F | F | T | T | F | F | F | F |
| F | F | F | F | F | F | F | F |
By comparing the truth values of both sides of the equation for each row, we can see that they are equal. Therefore, the statement verifies the Distributive Law.
2. Associative Law:
a) Statement: (P ∨ Q) ∨ R = P ∨ (P ∨ R)
Create a truth table for both sides of the equation:
| P | Q | R | P ∨ Q | (P ∨ Q) ∨ R | P ∨ R | P ∨ (P ∨ R) |
|---|---|---|-------|-------------|-------|-------------|
| T | T | T | T | T | T | T |
| T | T | F | T | T | T | T |
| T | F | T | T | T | T | T |
| T | F | F | T | T | T | T |
| F | T | T | T | T | T | T |
| F | T | F | T | T | F | F |
| F | F | T | F | T | T | F |
| F | F | F | F | F | F | F |
By comparing the truth values of both sides of the equation for each row, we can see that they are equal. Therefore, the statement verifies the Associative Law.
b) Statement: (P ∧ Q) ∧ R = P ∧ (Q ∧ R)
Create a truth table for both sides of the equation:
| P | Q | R | P ∧ Q | (P ∧ Q) ∧ R | Q ∧ R | P ∧ (Q ∧ R) |
|---|---|---|-------|-------------|-------|-------------|
| T | T | T | T | T | T | T |
| T | T | F | T | F | F | F |
| T | F | T | F | F | F | F |
| T | F | F | F | F | F | F |
| F | T | T | F | F | T | F |
| F | T | F | F | F | F | F |
| F | F | T | F | F | F | F |
| F | F | F | F | F | F | F |
By comparing the truth values of both sides of the equation for each row, we can see that they are equal. Therefore, the statement verifies the Associative Law.
In both cases, the truth tables confirm that the given statements are valid.