Prove that the propositional formulas given in the question i, ii, and iii above having the corresponding properties, by means of semantic tableau.

Question

i〉¬(p→q)↔(p∧¬q)

ii〉(p→q) ∧ (q→r) ∧¬ (p→r)

iii〉((p∧q)→r)→(p→(q∧r))

Prove that the propositional formulas given in the question i, ii, and iii above having the corresponding properties, by means of semantic tableau.

Summary

Here, they are given 3 proofs in the question it has some properties of semantic tableau form. In the below drawn a tree diagrams for the given questions. So, for these types of questions, we need to start the proofs because it is a backend process.

Explanation

i〉¬(p→q)↔(p∧¬q)

¬ (р->a) <-> (p^а)

/\

[¬(p->q)]^(p^¬q)     (p->q)^[¬(p^¬q)]

|                                 |

¬(p->q), (p^¬q)           (p->q), ¬(p^¬q)

|                                      /\

p, ¬q , (p^¬q)          ¬p, ¬(p^¬q)    q,¬(p^¬q)

|                             /\                       /\

p, ¬q, p, ¬q               ¬p, ¬p   ¬p, q      q, ¬p     q, p

|                       |           •              •            |

p, ¬q                    ¬p                                     q

•

 

ii). (p→q) ∧ (q→r) ∧¬ (p→r)

 

(p→q) ∧ (q→r) ∧¬ (p→r)

|

(p→q) , (q→r) ,¬ (p→r)

/\

¬p, (q→r) ,¬ (p→r)                q,(q→r),¬ (p→r)

/\                                                  /\

¬p, ¬q,¬ (p→r)     ¬p ,r,¬ (p→r)        q, ¬q,¬ (p→r)       q, r,¬ (p→r)

|                            |                         |                          |

¬p, ¬q, p, ¬r            ¬p ,r, p, ¬r           q,¬ q ,p, ¬r              q, r, p, ¬r

×                          ×                     ×                              ×

 

 

iii〉((p∧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           ¬ p     q ,r

 

 

Also, the read another blog which is to write a relational schema for the diagram.