Proving Equivalence using Truth Tables: p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r ), and p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r )

p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r )p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r )

To prove the first equation, p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r):

We will use the truth table method to prove the equivalence

To prove the first equation, p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r):

We will use the truth table method to prove the equivalence.

Truth Table for p ∧ (q ∨ r ):

| p | q | r | q ∨ r | p ∧ (q ∨ r ) |
|—|—|—|——-|————–|
| T | T | T | T | T |
| T | T | F | T | T |
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | F |
| F | T | F | T | F |
| F | F | T | T | F |
| F | F | F | F | F |

Truth Table for (p ∧ q) ∨ (p ∧ r ):

| p | q | r | p ∧ q | p ∧ r | (p ∧ q) ∨ (p ∧ r ) |
|—|—|—|——–|——-|——————|
| T | T | T | T | T | T |
| T | T | F | T | F | T |
| T | F | T | F | T | T |
| T | F | F | F | F | F |
| F | T | T | F | F | F |
| F | T | F | F | F | F |
| F | F | T | F | F | F |
| F | F | F | F | F | F |

By comparing the two truth tables, we can see that the values for p ∧ (q ∨ r ) and (p ∧ q) ∨ (p ∧ r ) are the same for all possible combinations of truth values for p, q, and r. Therefore, we can conclude that p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r ).

To prove the second equation, p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r ):

Again, we will use the truth table method.

Truth Table for p ∨ (q ∧ r ):

| p | q | r | q ∧ r | p ∨ (q ∧ r ) |
|—|—|—|——-|————–|
| T | T | T | T | T |
| T | T | F | F | T |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |

Truth Table for (p ∨ q) ∧ (p ∨ r ):

| p | q | r | p ∨ q | p ∨ 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 | T | F | F |
| F | T | T | T | T | T |
| F | T | F | T | F | F |
| F | F | T | F | T | F |
| F | F | F | F | F | F |

Again, by comparing the two truth tables, we can see that the values for p ∨ (q ∧ r ) and (p ∨ q) ∧ (p ∨ r ) are the same for all possible combinations of truth values for p, q, and r. Therefore, we can conclude that p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r ).

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