GATE CSE 1987 | Question: 10e

Question

Show that the conclusion $(r \to q)$ follows from the premises$:p, (p \to q) \vee (p \wedge (r \to q))$

Comments

In best answer it shows that it is tautology means it is valid(always true).

but assume,

p=true , r=True ,q=False

then it is False means it is invalid or not tautology.

anybody know what’s going on.??

EDIT : This will be a tautology read below comment.

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@ASNR1010  

(p ∧ (p→q)∨(p∧(r→q))) → (r→q)  should be tautology.

For the value p=true , r=True ,q=False.
It becomes 
(T ^ (F v (T ^(F)))) → F which becomes F → F.
which is a Tautology

Answer 1

p

(p → q) V (p ^ (r → q))

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r → q

 

For the conclusion (r → q) to hold true,

We will try to make the premises True and the Conclusion as False, 

If we are able to do so then T→ F = F which means the conclusion does not follow the premise.

 

Now, r→ q = F so r = T, q = F

And, p = T

(T → q) V (T ^ (r → q)) = q V (r → q) = q V (r’ V q) = q V r’ = F V F = F (must have been True)

 

So, we couldn’t make the premises True so this conclusion holds.