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)
So, what are your truth tables?
Sure! To verify these two laws using a truth table, we will need to consider all possible combinations of truth values for the propositions involved.
1. Distributive Law:
We have the expression: P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)
To construct the truth table, we need columns for P, Q, R, (Q ∨ R), (P ∧ Q), (P ∧ R), and the final expression (P ∧ (Q ∨ R)) = ((P ∧ Q) ∨ (P ∧ R)).
The truth table would have 2^3 = 8 rows, representing all the possible combinations of truth values for P, Q, and R.
Here is the truth table:
| P | Q | R | (Q ∨ R) | (P ∧ Q) | (P ∧ R) | P ∧ (Q ∨ R) | ((P ∧ Q) ∨ (P ∧ R)) |
|---|---|---|---------|---------|---------|-------------|--------------------|
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | F | T | T |
| T | F | T | T | F | T | 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 |
The last two columns in the truth table (P ∧ (Q ∨ R)) and ((P ∧ Q) ∨ (P ∧ R)) have the same values for all rows. Hence, the Distributive Law is verified.
2. Associative Law:
a. For the expression: (P ∨ Q) ∨ R = P ∨ (P ∨ R)
Constructing the truth table, we have columns for P, Q, R, (P ∨ Q), (P ∨ R), and the final expression (P ∨ Q) ∨ R = P ∨ (P ∨ R).
Again, the truth table would have 2^3 = 8 rows.
Here is the truth table:
| P | Q | R | (P ∨ Q) | (P ∨ R) | (P ∨ Q) ∨ 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 | F | T | F |
| F | F | T | F | T | T | T |
| F | F | F | F | F | F | F |
The last two columns in the truth table (P ∨ Q) ∨ R and P ∨ (P ∨ R) have the same values for all rows. Therefore, the Associative Law is verified for disjunction (or).
b. For the expression: (P ∧ Q) ∧ R = P ∧ (Q ∧ R)
Constructing the truth table, we have columns for P, Q, R, (P ∧ Q), (Q ∧ R), and the final expression (P ∧ Q) ∧ R = P ∧ (Q ∧ R).
Using 2^3 = 8 rows again:
| P | Q | R | (P ∧ Q) | (Q ∧ R) | (P ∧ 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 | T | F | F |
| F | T | F | F | F | F | F |
| F | F | T | F | F | F | F |
| F | F | F | F | F | F | F |
The last two columns in the truth table (P ∧ Q) ∧ R and P ∧ (Q ∧ R) have the same values for all rows. Therefore, the Associative Law is verified for conjunction (and).