Conjunction and operation of propositions

If you can't see clearly, you can look at the following website (resources)

3. Find the principal disjunctive normal form and principal conjunctive normal form of the formula P ∧ Q ∨ R.

P ∧ Q ∨ R ( P ∧ Q )∨ R

1 truth table method

P Q R (P∧Q)∨R minimax term

0 0 0 0 P∨Q∨R

0 0 1 1 ┐P∧Q∧R

0 1 0 0 P∨┐Q∨R

┐P∧Q∧R

1 0 0 0 ┐P∨Q∨R

1 0 1 1 P∧┐Q∧R

1 1 0 1 P∧Q∧┐R

1 1 1 1 P∧Q∧R

(P ∧ Q )∨ R

(┐p∧┐q∧r)∨(┐p∧q∧r)∨( p∧┐q∧r)∨( p∧q∧┐r)∨( p∧q∧r)

Principal disjunctive paradigm;

(P ∧ Q )∨ R

< ==>( P ∨ Q ∨ R )∧( P ∨┐ Q ∨ R )∧(┐ P ∨ Q ∨ R)

Conjunctive normal form is preferred.

2 equivalent algorithm

(P ∧ Q )∨ R

< = = >(p∧q∧(r∨┐r))∨((p∨┐p)∧(q∨┐q)∧r)

< = = >(p∧q∧r)∨( p∧q∧┐r)∨( p∧┐q∧r)∨(┐p∧q∧r)∨(┐p∧┐q∧r)

< = = >(┐p∧┐q∧r)∨(┐p∧q∧r)∨( p∧q∧r)∨。

The abbreviation is ∑( 1, 3,5,6,7).

(P ∧ Q )∨ R ( P ∨ R )∧( Q ∨ R)

< ==>( P ∨( Q ∧┐ Q )∨ R )∧(( P ∧┐ P )∨ Q ∨ R)

< = = >(p∨q∨r)∧(p∨┐q∨r)∧(p∨q∨r )∧(┐p q∨r)

< ==>( P ∨ Q ∨ R )∧( P ∨┐ Q ∨ R )∧(┐ P ∨ Q ∨ R)

The abbreviation (or code) of (main conjunctive normal form) is ∏ (0,2,4).