How to use semantic tableaux to prove validity of the formula, $(P \wedge Q) \to (P \wedge (P \to Q))$

Question

I have made an attempt to prove this formulas validity using semantic tableaux, but I'm not sure that I'm doing it correctly. Here is my attempt.

$(P \wedge Q) \to (P \wedge (P\to Q))$ is valid.

1. $\neg ((P \wedge Q) \to (P \wedge (P\to Q)))$
2. $P\wedge Q$
3. $\neg (P \wedge (P \to Q))$
4. $P$
5. $Q$
6. $\neg P$

X

7. $\neg (P→Q)$
8. $P$
9. $\neg Q$

X

Would anyone be able to correct me on this or let me know if its correct.

Thank you in advance.

Answer 1

To prove: $P\land Q\implies P\land (P\implies Q)$

It is best to use conditional proof or proof by contradiction.

The tableaux method is very similar to my understanding of Patric Suppes's book, 'Introduction to logic$ way of derivation.

Here goes: Assume the consequent is false:

$1. \neg (P\land(P\implies Q))$

$2. \neg P \lor \neg(P\implies Q), 1 T$

$3. \neg (P\implies Q)\iff \neg (\neg P\lor Q)$ Tautology of implication

$4. \neg(\neg P\lor Q)\iff P\land \neg Q$

$5. \neg P\lor(P\land \neg Q)\iff (\neg P\lor P) \land (\neg P\lor \neg Q)$

$6. \neg P\lor\neg Q\iff \neg(P\land Q)$

$7. \neg(P\land(P\implies Q))\implies \neg(P\land Q), 1, 6 T$

$8. P\land Q\implies P\land (P\implies Q), 7T$

The $nT$ symbols at end of formulae stand for line number Tautology. i.e. $7T$ implies $8$ is tautologically equivalent to $7$.

Answer 2

I would simply do it, going over all possible values:

P ^ Q -> (P ^ (P -> Q))
=======================
0 0 0  1  0 0  0  1 0
0 0 1  1  0 0  0  1 1
1 0 0  1  1 0  1  0 0
1 1 1  1  1 1  1  1 1